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            Lecture 21: Random Variables
         
    
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<p><strong>Description:</strong> Introduces partitioning of the probabilistic sample space using random variables. Distribution functions, notably, the binomial distribution, are discussed.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
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<p><span m="400">The</span> <span m="510">following</span> <span m="950">content</span> <span m="1550">is</span> <span m="1660">provided</span> <span m="2110">under</span> <span m="2390">a</span> <span m="2430">Creative</span> <span m="2830">Commons</span> <span m="3230">license.</span> <span m="4340">Your</span> <span m="4530">support</span> <span m="5040">will</span> <span m="5200">help</span> <span m="5440">MIT</span> <span m="5890">OpenCourseWare</span> <span m="6680">continue</span> <span m="7190">to</span> <span m="7270">offer</span> <span m="7690">high</span> <span m="7930">quality</span> <span m="8440">educational</span> <span m="9070">resources</span> <span m="9700">for</span> <span m="9850">free.</span> <span m="11050">To</span> <span m="11060">make</span> <span m="11260">a</span> <span m="11310">donation,</span> <span m="11990">or</span> <span m="12260">view</span> <span m="12700">additional</span> <span m="13130">materials</span> <span m="13660">from</span> <span m="13810">hundreds</span> <span m="14250">of</span> <span m="14360">MIT</span> <span m="14780">courses,</span> <span m="15900">visit</span> <span m="16110">MIT</span> <span m="16530">OpenCourseWare</span> <span m="17580">at</span> <span m="17740">ocw.mit.edu.</span> </p>
<p><span m="24160">PROFESSOR: So</span> <span m="24390">far</span> <span m="24620">in</span> <span m="24700">our</span> <span m="24790">discussion</span> <span m="25270">of</span> <span m="25340">probability,</span> <span m="26400">we've</span> <span m="26630">focused</span> <span m="27810">our</span> <span m="27990">attention</span> <span m="28490">on</span> <span m="28570">the</span> <span m="28670">probability</span> <span m="29210">that</span> <span m="29370">some</span> <span m="29750">kind</span> <span m="30290">of</span> <span m="30450">event</span> <span m="30940">occurs--</span> <span m="32350">the</span> <span m="32470">probability</span> <span m="33100">you</span> <span m="33220">win</span> <span m="33570">the</span> <span m="33650">Monty</span> <span m="33990">Hall</span> <span m="34280">game,</span> <span m="35370">the</span> <span m="35380">probability</span> <span m="36120">that</span> <span m="36310">you</span> <span m="36430">get</span> <span m="36600">a</span> <span m="36680">heads</span> <span m="37170">when</span> <span m="37310">you</span> <span m="37410">flip</span> <span m="37640">a</span> <span m="37720">coin,</span> <span m="39390">the</span> <span m="39700">probability</span> <span m="40290">we</span> <span m="40490">all</span> <span m="40870">have</span> <span m="41120">different</span> <span m="41440">birthdays</span> <span m="42120">in</span> <span m="42200">the</span> <span m="42320">room--</span> <span m="43190">that</span> <span m="43400">kind</span> <span m="43590">of</span> <span m="43700">thing.</span> <span m="44570">In</span> <span m="44690">each</span> <span m="44920">case,</span> <span m="45500">the</span> <span m="45550">event</span> <span m="45900">happens</span> <span m="46800">or</span> <span m="47020">it</span> <span m="47140">doesn't.</span> <span m="47760">It's</span> <span m="47920">a</span> <span m="47970">zero-one</span> <span m="48610">kind</span> <span m="48820">of</span> <span m="48910">thing.</span> </p>
<p><span m="50630">For</span> <span m="50780">the</span> <span m="50910">rest</span> <span m="51210">of</span> <span m="51290">the</span> <span m="52090">course,</span> <span m="52690">we're</span> <span m="52710">going</span> <span m="52790">to</span> <span m="52850">start</span> <span m="53210">talking</span> <span m="53490">about</span> <span m="53690">more</span> <span m="53850">complicated</span> <span m="54650">situations.</span> <span m="56460">Instead</span> <span m="56760">of</span> <span m="56810">talking</span> <span m="57140">about</span> <span m="57570">zero-one</span> <span m="58140">events,</span> <span m="58840">like</span> <span m="59060">winning</span> <span m="59360">or</span> <span m="59420">losing,</span> <span m="60490">we're</span> <span m="60800">going</span> <span m="60890">to</span> <span m="60950">start</span> <span m="61370">talking</span> <span m="61700">about</span> <span m="61980">how</span> <span m="62250">much</span> <span m="63220">you</span> <span m="63420">win.</span> <span m="64610">Instead</span> <span m="65000">of</span> <span m="65379">talking</span> <span m="65770">about</span> <span m="66590">whether</span> <span m="66860">or not you</span> <span m="66960">get</span> <span m="67100">a</span> <span m="67150">heads</span> <span m="67450">in</span> <span m="67550">a</span> <span m="67620">coin</span> <span m="67980">flip,</span> <span m="68730">we're</span> <span m="68970">going</span> <span m="69060">to</span> <span m="69140">talk</span> <span m="69400">about</span> <span m="69660">flipping</span> <span m="69960">lots</span> <span m="70300">of</span> <span m="70420">coins</span> <span m="70930">and</span> <span m="71040">asking</span> <span m="71780">how</span> <span m="72010">many</span> <span m="72380">heads</span> <span m="73340">did</span> <span m="73510">you</span> <span m="73620">get?</span> </p>
<p><span m="76080">Now,</span> <span m="76970">this</span> <span m="77380">involves</span> <span m="78860">the</span> <span m="79040">notion</span> <span m="79480">of</span> <span m="79660">a</span> <span m="79960">random</span> <span m="80520">variable</span> <span m="81420">to</span> <span m="81540">count</span> <span m="81920">how</span> <span m="82140">much</span> <span m="82690">or</span> <span m="82870">how</span> <span m="83020">many</span> <span m="83530">in</span> <span m="83650">a</span> <span m="83700">random</span> <span m="83970">situation.</span> <span m="84990">And</span> <span m="85130">actually,</span> <span m="85380">the</span> <span m="85470">name</span> <span m="85830">is</span> <span m="85940">sort</span> <span m="86230">of</span> <span m="86380">weird.</span> <span m="87250">It's</span> <span m="87480">not</span> <span m="87690">a</span> <span m="87730">variable</span> <span m="88480">at</span> <span m="88620">all.</span> <span m="89650">In</span> <span m="89680">fact,</span> <span m="90140">a</span> <span m="90230">random</span> <span m="90600">variable</span> <span m="91110">is</span> <span m="91220">a</span> <span m="91290">function</span> <span m="92370">from</span> <span m="92540">the</span> <span m="92630">sample</span> <span m="93080">space</span> <span m="93500">to</span> <span m="93580">the</span> <span m="93690">real</span> <span m="93910">numbers.</span> </p>
<p><span m="105390">See</span> <span m="105980">random</span> <span m="106380">variable</span> <span m="106870">R</span> <span m="108690">is</span> <span m="108920">a</span> <span m="109090">function</span> <span m="114520">R</span> <span m="115080">from</span> <span m="115630">the</span> <span m="115740">sample</span> <span m="116220">space</span> <span m="117880">to</span> <span m="118020">the</span> <span m="118170">reals.</span> <span m="120660">So</span> <span m="120730">this</span> <span m="120940">is</span> <span m="121050">the</span> <span m="121160">random</span> <span m="121510">variable,</span> <span m="121980">and we'll</span> <span m="122130">often</span> <span m="122470">denote</span> <span m="122780">that</span> <span m="122970">by</span> <span m="123420">rv.</span> <span m="125040">This</span> <span m="125330">is</span> <span m="125470">the</span> <span m="125580">sample</span> <span m="126010">space</span> <span m="126830">of</span> <span m="126980">all</span> <span m="127130">possible</span> <span m="127650">outcomes.</span> <span m="131190">And</span> <span m="131360">this</span> <span m="131530">is</span> <span m="131610">just</span> <span m="131870">the</span> <span m="131990">reals.</span> </p>
<p><span m="137460">All</span> <span m="137520">right,</span> <span m="137680">so</span> <span m="137830">in</span> <span m="137910">other</span> <span m="138080">words,</span> <span m="138700">a</span> <span m="138800">random</span> <span m="139150">variable</span> <span m="139600">is</span> <span m="139730">just</span> <span m="140070">a</span> <span m="140140">way</span> <span m="140500">of</span> <span m="140670">assigning</span> <span m="141250">a</span> <span m="141390">real</span> <span m="141760">number</span> <span m="142150">to</span> <span m="142540">each</span> <span m="142900">possible</span> <span m="143510">outcome.</span> <span m="145440">For</span> <span m="145570">example,</span> <span m="146170">say</span> <span m="146400">that</span> <span m="146580">we</span> <span m="146860">toss</span> <span m="147430">three</span> <span m="147720">coins.</span> <span m="156560">We</span> <span m="156960">could</span> <span m="157190">let</span> <span m="157480">R</span> <span m="159020">equal</span> <span m="159410">the</span> <span m="159520">number</span> <span m="159880">of</span> <span m="159960">heads</span> <span m="163270">among</span> <span m="163560">the</span> <span m="163650">three</span> <span m="163880">coins.</span> <span m="165080">And</span> <span m="165370">R</span> <span m="165680">is</span> <span m="165770">thus</span> <span m="166000">a</span> <span m="166090">random</span> <span m="166420">variable.</span> <span m="167040">For</span> <span m="167080">example,</span> <span m="169450">for</span> <span m="169550">the</span> <span m="169900">outcome</span> <span m="170520">where</span> <span m="170660">the</span> <span m="170770">first</span> <span m="171050">coin is</span> <span m="171400">heads,</span> <span m="171920">the</span> <span m="172020">second</span> <span m="172410">is</span> <span m="172530">tails,</span> <span m="173000">and the</span> <span m="173110">third</span> <span m="173360">is</span> <span m="173480">heads,</span> <span m="174050">R</span> <span m="174530">would</span> <span m="174980">be</span> <span m="175890">the</span> <span m="176000">real</span> <span m="176180">number</span> <span m="176480">2,</span> <span m="177010">indicating</span> <span m="177570">I</span> <span m="177610">got</span> <span m="178030">two</span> <span m="178280">heads</span> <span m="178650">in</span> <span m="178740">that</span> <span m="178920">outcome.</span> </p>
<p><span m="182160">We</span> <span m="182290">could</span> <span m="182410">also</span> <span m="182670">define</span> <span m="183030">another</span> <span m="184230">value</span> <span m="184600">called</span> <span m="184970">M,</span> <span m="186510">which</span> <span m="186830">we'll</span> <span m="186980">say</span> <span m="187300">is</span> <span m="187480">1</span> <span m="189450">if</span> <span m="189690">all</span> <span m="189980">three</span> <span m="190230">coins</span> <span m="190620">match,</span> <span m="193440">if they're</span> <span m="193600">all</span> <span m="193750">the</span> <span m="193850">same,</span> <span m="196680">and</span> <span m="196960">0</span> <span m="198350">otherwise,</span> <span m="198910">if</span> <span m="199010">they're</span> <span m="199100">not</span> <span m="199330">all</span> <span m="199500">the same.</span> <span m="203400">For</span> <span m="203650">example,</span> <span m="206160">M</span> <span m="206560">on</span> <span m="206730">the</span> <span m="206900">outcome</span> <span m="207610">heads,</span> <span m="208660">heads,</span> <span m="209270">tails</span> <span m="210385">is</span> <span m="210750">0.</span> <span m="212780">M</span> <span m="213160">on</span> <span m="213330">the</span> <span m="213510">outcome</span> <span m="214550">tails,</span> <span m="215110">tails,</span> <span m="215650">tails</span> <span m="216190">would</span> <span m="216330">be</span> <span m="216430">1,</span> <span m="216920">because</span> <span m="217120">they</span> <span m="217260">all</span> <span m="217610">are</span> <span m="217730">the</span> <span m="217810">same.</span> </p>
<p><span m="220930">Is</span> <span m="221150">M a</span> <span m="221480">random</span> <span m="221810">variable?</span> <span m="225480">Yeah,</span> <span m="226480">it</span> <span m="226600">assigns</span> <span m="227060">a</span> <span m="227100">value,</span> <span m="227680">a</span> <span m="227790">real</span> <span m="227990">number,</span> <span m="228290">to</span> <span m="228390">every</span> <span m="228730">outcome.</span> <span m="229370">And</span> <span m="229490">that</span> <span m="229670">value</span> <span m="230030">is</span> <span m="230130">either</span> <span m="230350">0</span> <span m="231040">or</span> <span m="231180">1,</span> <span m="231710">depending</span> <span m="232110">on</span> <span m="232210">if</span> <span m="232320">all</span> <span m="232500">the</span> <span m="232590">coins</span> <span m="232930">match.</span> <span m="233980">In</span> <span m="234100">fact,</span> <span m="234340">it's</span> <span m="234470">a</span> <span m="234500">very</span> <span m="234720">special</span> <span m="235210">kind</span> <span m="235480">of</span> <span m="235570">random</span> <span m="235910">variable</span> <span m="236870">that's</span> <span m="237030">called</span> <span m="237300">an</span> <span m="237440">indicator</span> <span m="238450">random</span> <span m="238760">variable,</span> <span m="239250">also</span> <span m="239520">known</span> <span m="239720">as</span> <span m="239880">Bernoulli</span> <span m="240840">or</span> <span m="241050">characteristic</span> <span m="242110">random</span> <span m="242450">variable,</span> <span m="243820">because</span> <span m="244570">it's</span> <span m="244930">zero-one.</span> </p>
<p><span m="248860">An</span> <span m="249050">indicator</span> <span m="253770">also</span> <span m="254220">known</span> <span m="254510">as</span> <span m="256019">a</span> <span m="256250">Bernoulli</span> <span m="261209">or</span> <span m="261579">characteristic</span> <span m="272310">random</span> <span m="272660">variable,</span> <span m="275890">is</span> <span m="276200">a</span> <span m="276320">random</span> <span m="276740">variable</span> <span m="279120">whose</span> <span m="279380">range</span> <span m="282480">or</span> <span m="282620">possible</span> <span m="283080">values</span> <span m="284130">is</span> <span m="284350">just</span> <span m="284610">0</span> <span m="284980">and</span> <span m="285100">1.</span> <span m="287910">So</span> <span m="288200">it's</span> <span m="288320">a</span> <span m="288370">random</span> <span m="288630">variable</span> <span m="289040">where</span> <span m="289280">the</span> <span m="289430">real</span> <span m="289630">numbers</span> <span m="290030">are</span> <span m="290160">0,</span> <span m="290500">1</span> <span m="290730">that you</span> <span m="290860">assign to</span> <span m="291270">every</span> <span m="291480">sample</span> <span m="291860">point.</span> <span m="293140">And</span> <span m="293340">it's</span> <span m="294300">called</span> <span m="294640">a</span> <span m="294690">characteristic</span> <span m="295580">or</span> <span m="295690">indicator</span> <span m="296260">random</span> <span m="296530">variable</span> <span m="296920">because</span> <span m="297880">it</span> <span m="298150">indicates</span> <span m="299120">which</span> <span m="299550">sample</span> <span m="300060">points</span> <span m="300500">have</span> <span m="300660">a</span> <span m="300720">certain</span> <span m="301060">property.</span> </p>
<p><span m="302430">In</span> <span m="302560">this</span> <span m="302760">case,</span> <span m="303920">all</span> <span m="304260">the</span> <span m="304380">coins</span> <span m="304740">were</span> <span m="304800">the</span> <span m="304910">same.</span> <span m="305480">Or it</span> <span m="305950">indicates</span> <span m="306600">that a</span> <span m="306770">sample</span> <span m="307100">point</span> <span m="307310">has</span> <span m="307430">a</span> <span m="307480">certain</span> <span m="307710">characteristic,</span> <span m="308940">which</span> <span m="309110">is</span> <span m="309160">why</span> <span m="309350">it's</span> <span m="309440">called</span> <span m="309670">a</span> <span m="309710">characteristic</span> <span m="310430">random</span> <span m="310680">variable.</span> <span m="311780">And</span> <span m="312000">we're</span> <span m="312130">going</span> <span m="312210">to</span> <span m="312250">be</span> <span m="312360">talking</span> <span m="312730">a</span> <span m="312800">lot</span> <span m="313200">about</span> <span m="313560">these</span> <span m="313750">kinds</span> <span m="313990">of</span> <span m="314070">random</span> <span m="314340">variables</span> <span m="314800">over</span> <span m="314950">the</span> <span m="315030">next</span> <span m="315240">couple</span> <span m="315480">of</span> <span m="315570">weeks.</span> </p>
<p><span m="317760">Now,</span> <span m="317900">in</span> <span m="318020">general,</span> <span m="318800">a</span> <span m="318920">random</span> <span m="319240">variable</span> <span m="320280">is</span> <span m="320450">equivalent</span> <span m="321110">to,</span> <span m="321440">or</span> <span m="321590">it</span> <span m="321920">can</span> <span m="322100">define</span> <span m="322670">a</span> <span m="322740">partition</span> <span m="323820">on</span> <span m="323940">the</span> <span m="324020">sample</span> <span m="324430">space.</span> <span m="326470">For</span> <span m="326610">an</span> <span m="326670">indicator</span> <span m="327180">random</span> <span m="327490">variable,</span> <span m="329710">it</span> <span m="329920">defines</span> <span m="330540">a</span> <span m="330620">partition</span> <span m="331370">with</span> <span m="331560">two</span> <span m="331780">blocks.</span> <span m="333220">For</span> <span m="333300">example,</span> <span m="334790">say</span> <span m="334940">we</span> <span m="335120">look</span> <span m="335280">at</span> <span m="335360">the</span> <span m="335450">sample</span> <span m="335850">space</span> <span m="336380">with</span> <span m="337490">three</span> <span m="338070">coin</span> <span m="338470">tosses</span> <span m="339020">there.</span> <span m="341080">There's</span> <span m="341320">eight</span> <span m="341610">outcomes.</span> <span m="342180">There's</span> <span m="342400">heads,</span> <span m="342710">heads,</span> <span m="343080">heads;</span> <span m="346120">heads,</span> <span m="347590">tails,</span> <span m="348150">heads;</span> <span m="350400">heads,</span> <span m="350790">heads,</span> <span m="351240">tails;</span> <span m="353150">tail,</span> <span m="353840">heads,</span> <span m="354240">heads;</span> <span m="355556">and</span> <span m="355950">the</span> <span m="356030">reverse</span> <span m="356470">of</span> <span m="356550">those.</span> </p>
<p><span m="364470">This</span> <span m="364690">is</span> <span m="364840">the</span> <span m="364910">sample</span> <span m="365330">space</span> <span m="365820">when</span> <span m="365970">I</span> <span m="366030">toss</span> <span m="366410">three</span> <span m="366640">coins.</span> <span m="367180">There's</span> <span m="367440">eight</span> <span m="367790">possible</span> <span m="368280">outcomes.</span> <span m="369940">The</span> <span m="370080">random</span> <span m="370450">variable</span> <span m="370940">M</span> <span m="371890">defines</span> <span m="372360">this</span> <span m="372550">partition.</span> <span m="375700">Up</span> <span m="375890">top,</span> <span m="376200">you</span> <span m="376300">have</span> <span m="376490">M</span> <span m="376670">equals</span> <span m="376970">1,</span> <span m="377385">and down</span> <span m="377800">below</span> <span m="378070">you</span> <span m="378170">have</span> <span m="378390">M</span> <span m="378600">equals</span> <span m="378900">0.</span> <span m="380790">Because</span> <span m="380990">these</span> <span m="381240">are</span> <span m="381440">the</span> <span m="381660">outcomes</span> <span m="382120">where</span> <span m="382250">all</span> <span m="382470">the</span> <span m="382540">coins</span> <span m="382830">match.</span> <span m="383560">These</span> <span m="383810">are the</span> <span m="383910">ones</span> <span m="384150">that</span> <span m="384270">don't.</span> </p>
<p><span m="387820">The</span> <span m="388060">random</span> <span m="388430">variable</span> <span m="388880">R</span> <span m="389330">defines</span> <span m="390000">a</span> <span m="390100">different</span> <span m="390420">partition.</span> <span m="393740">In</span> <span m="393840">that</span> <span m="394050">case,</span> <span m="394510">the</span> <span m="394610">partition</span> <span m="395090">has</span> <span m="395310">four</span> <span m="395610">blocks.</span> <span m="398320">Here</span> <span m="398760">is</span> <span m="400410">R</span> <span m="400680">equals</span> <span m="401000">0,</span> <span m="402420">no</span> <span m="402640">heads.</span> <span m="404860">This</span> <span m="405060">is</span> <span m="405200">R</span> <span m="405420">equals</span> <span m="405730">1,</span> <span m="406150">to</span> <span m="406540">one</span> <span m="406760">head.</span> <span m="408660">R</span> <span m="408940">equals</span> <span m="409280">2,</span> <span m="410210">and</span> <span m="410560">R</span> <span m="410770">equals</span> <span m="411110">3.</span> <span m="414460">So</span> <span m="414580">a</span> <span m="414650">random</span> <span m="415000">variable</span> <span m="415480">just</span> <span m="415680">sort of</span> <span m="415840">organizes</span> <span m="416520">your</span> <span m="416690">sample</span> <span m="417060">space,</span> <span m="418750">partitions</span> <span m="419180">it</span> <span m="419340">into</span> <span m="419430">blocks,</span> <span m="420980">defined</span> <span m="421450">by</span> <span m="422310">all</span> <span m="422570">the</span> <span m="422690">elements</span> <span m="423100">in</span> <span m="423140">the</span> <span m="423200">block,</span> <span m="423560">the</span> <span m="423670">outcomes</span> <span m="424030">in</span> <span m="424080">the</span> <span m="424140">block</span> <span m="424550">have</span> <span m="424700">the</span> <span m="424780">same</span> <span m="425260">value</span> <span m="426200">for</span> <span m="426280">the</span> <span m="426400">random</span> <span m="426700">variable.</span> </p>
<p><span m="429500">And,</span> <span m="429670">in</span> <span m="429780">fact,</span> <span m="430400">every</span> <span m="430560">block</span> <span m="431100">is</span> <span m="431260">really</span> <span m="431710">an</span> <span m="431800">event.</span> <span m="434000">In</span> <span m="434130">particular,</span> <span m="434770">we</span> <span m="434780">can</span> <span m="434960">say</span> <span m="435250">that,</span> <span m="436790">if</span> <span m="436920">you</span> <span m="437040">look</span> <span m="437220">at</span> <span m="437280">the</span> <span m="437440">outcomes</span> <span m="438680">for</span> <span m="438950">which</span> <span m="440600">the</span> <span m="440760">random</span> <span m="441110">variable</span> <span m="441530">on</span> <span m="441640">that</span> <span m="441810">outcome</span> <span m="442220">equals</span> <span m="442490">some</span> <span m="442700">value</span> <span m="443050">x,</span> <span m="444830">this</span> <span m="445050">is</span> <span m="445180">simply</span> <span m="445570">the</span> <span m="445730">event</span> <span m="450130">that</span> <span m="450360">R</span> <span m="450580">equals</span> <span m="450860">x.</span> <span m="452160">The</span> <span m="452260">random</span> <span m="452580">variable</span> <span m="453520">is</span> <span m="453700">x.</span> <span m="454900">And,</span> <span m="454980">of</span> <span m="455020">course,</span> <span m="455270">an</span> <span m="455370">event</span> <span m="455760">is</span> <span m="455860">just</span> <span m="456080">a</span> <span m="456140">subset</span> <span m="456650">of</span> <span m="456710">the</span> <span m="456800">sample</span> <span m="457160">space.</span> <span m="457490">It's</span> <span m="457620">a</span> <span m="457670">collection</span> <span m="458110">of</span> <span m="458170">outcomes</span> <span m="459410">that</span> <span m="459540">define</span> <span m="459930">the</span> <span m="460040">event.</span> </p>
<p><span m="462890">And</span> <span m="463050">since</span> <span m="463210">we</span> <span m="463330">know</span> <span m="463840">we</span> <span m="463980">could</span> <span m="464100">talk</span> <span m="464310">about</span> <span m="464610">probabilities</span> <span m="465280">of</span> <span m="465330">events,</span> <span m="465870">we</span> <span m="466000">can</span> <span m="466130">now</span> <span m="466650">say</span> <span m="466950">things</span> <span m="467240">like</span> <span m="467440">this.</span> <span m="467900">We</span> <span m="467930">can</span> <span m="468070">say</span> <span m="470440">the</span> <span m="470540">probability</span> <span m="471240">that</span> <span m="471360">a</span> <span m="471430">random</span> <span m="471760">variable</span> <span m="472160">equals</span> <span m="472610">x</span> <span m="474280">is</span> <span m="474420">simply</span> <span m="474770">the</span> <span m="474890">probability</span> <span m="475680">of</span> <span m="475840">that</span> <span m="476040">event</span> <span m="476430">happening,</span> <span m="477800">which</span> <span m="477950">is</span> <span m="478090">the</span> <span m="478210">sum</span> <span m="478670">over</span> <span m="478910">all</span> <span m="479150">outcomes</span> <span m="480260">for</span> <span m="480510">which</span> <span m="482020">the</span> <span m="482120">random</span> <span m="482400">variable</span> <span m="483890">equals</span> <span m="484270">x for</span> <span m="484550">that</span> <span m="484650">outcome,</span> <span m="486080">summing</span> <span m="486540">its</span> <span m="486660">probability.</span> <span m="488850">So</span> <span m="488980">the</span> <span m="489090">probability</span> <span m="489670">that</span> <span m="490450">the</span> <span m="490550">random</span> <span m="490770">variable</span> <span m="491120">equals</span> <span m="491460">a</span> <span m="491510">value</span> <span m="492060">is</span> <span m="492140">the</span> <span m="492240">sum</span> <span m="492640">of</span> <span m="492700">the</span> <span m="492790">probabilities</span> <span m="493490">of</span> <span m="493570">the</span> <span m="493660">sample</span> <span m="494100">points</span> <span m="494500">for</span> <span m="494620">which</span> <span m="495150">R</span> <span m="495420">has</span> <span m="495690">that</span> <span m="495940">value.</span> </p>
<p><span m="497580">All right,</span> <span m="497680">this</span> <span m="497820">is</span> <span m="497930">all</span> <span m="498130">really</span> <span m="498490">basic</span> <span m="498900">stuff,</span> <span m="499250">but</span> <span m="500720">important</span> <span m="501120">to</span> <span m="501190">get</span> <span m="501400">down.</span> <span m="503200">So,</span> <span m="503430">for</span> <span m="503560">example,</span> <span m="504190">what's</span> <span m="504340">the</span> <span m="504440">probability</span> <span m="506960">that</span> <span m="507090">R</span> <span m="507430">equals</span> <span m="507960">2</span> <span m="509060">in</span> <span m="509200">our</span> <span m="509310">case</span> <span m="509630">with</span> <span m="509760">the</span> <span m="509820">three</span> <span m="510060">coins?</span> <span m="511372">You</span> <span m="511760">flip</span> <span m="512030">three</span> <span m="512210">coins--</span> <span m="512480">let's</span> <span m="512650">say</span> <span m="512740">the</span> <span m="512850">coins</span> <span m="513190">are</span> <span m="513250">fair</span> <span m="514370">and</span> <span m="514620">mutually</span> <span m="515090">independent.</span> <span m="515860">We</span> <span m="515990">probably</span> <span m="516260">need</span> <span m="516409">to</span> <span m="516500">know</span> <span m="516640">that.</span> <span m="518720">What</span> <span m="519080">is</span> <span m="519240">the</span> <span m="519330">probability</span> <span m="519870">R</span> <span m="520120">equals</span> <span m="520400">2?</span> </p>
<p><span m="521675">AUDIENCE: 3/8?</span> </p>
<p><span m="522950">PROFESSOR: 3/8,</span> <span m="523909">because</span> <span m="524360">you've</span> <span m="524530">got</span> <span m="526120">three</span> <span m="526790">outcomes,</span> <span m="527510">each</span> <span m="527780">with a</span> <span m="527820">probability</span> <span m="528320">1/8.</span> <span m="530050">All right,</span> <span m="530460">so</span> <span m="530620">by</span> <span m="530740">this</span> <span m="530950">definition,</span> <span m="532390">that</span> <span m="532590">would</span> <span m="532740">be</span> <span m="533900">the</span> <span m="534010">same</span> <span m="534300">as</span> <span m="534410">the</span> <span m="534500">probability</span> <span m="535180">of</span> <span m="535900">heads,</span> <span m="536250">heads,</span> <span m="536640">tails;</span> <span m="537837">heads,</span> <span m="538631">tails,</span> <span m="539030">heads;</span> <span m="540490">tails,</span> <span m="540840">heads,</span> <span m="541240">heads;</span> <span m="542480">that's</span> <span m="542830">3/8.</span> </p>
<p><span m="546760">What's</span> <span m="547160">the</span> <span m="547270">probability</span> <span m="548660">that</span> <span m="548890">M</span> <span m="549770">equals</span> <span m="550160">1?</span> <span m="556928">What's</span> <span m="557370">the</span> <span m="557460">probability</span> <span m="558150">that</span> <span m="558340">M</span> <span m="558680">equals</span> <span m="559030">1?</span> <span m="559300">M</span> <span m="559570">is</span> <span m="559820">1</span> <span m="560250">if</span> <span m="560620">all</span> <span m="560850">the</span> <span m="560930">coins</span> <span m="561260">match.</span> <span m="563050">1/4</span> <span m="563820">because</span> <span m="564470">you've</span> <span m="564570">got</span> <span m="565420">two</span> <span m="565770">cases</span> <span m="566270">there--</span> <span m="566770">heads,</span> <span m="567080">heads,</span> <span m="567360">heads--</span> <span m="568912">whoops--</span> <span m="570860">or</span> <span m="570960">tails,</span> <span m="571360">tails,</span> <span m="571720">tails;</span> <span m="572730">and</span> <span m="573030">that's</span> <span m="573750">2/8,</span> <span m="574270">or</span> <span m="574490">1/4.</span> </p>
<p><span m="576982">Now</span> <span m="577440">we</span> <span m="577570">can</span> <span m="577710">also</span> <span m="577990">talk</span> <span m="578270">about</span> <span m="579100">the</span> <span m="579200">probability</span> <span m="579840">that</span> <span m="579970">a</span> <span m="580070">random</span> <span m="580490">variable</span> <span m="580970">is</span> <span m="581090">in</span> <span m="581260">some</span> <span m="581650">range.</span> <span m="583930">For</span> <span m="584060">example,</span> <span m="584630">I</span> <span m="584730">could</span> <span m="584760">ask,</span> <span m="585250">what's</span> <span m="585370">the</span> <span m="585470">probability</span> <span m="587180">that</span> <span m="587390">R</span> <span m="588290">is</span> <span m="589020">at</span> <span m="589220">least</span> <span m="589650">2?</span> <span m="592525">All right,</span> <span m="592990">that</span> <span m="593200">would</span> <span m="593410">mean</span> <span m="594930">the</span> <span m="595040">sum</span> <span m="595570">i</span> <span m="595880">equals</span> <span m="596290">2</span> <span m="596580">to</span> <span m="596690">infinity,</span> <span m="598340">the</span> <span m="598450">probability</span> <span m="599170">R</span> <span m="599540">equals</span> <span m="599930">i.</span> <span m="602230">And</span> <span m="602510">what</span> <span m="602660">is</span> <span m="602760">that?</span> <span m="605930">What's</span> <span m="606160">the</span> <span m="606250">probability?</span> <span m="607150">R is</span> <span m="607500">at</span> <span m="607580">least</span> <span m="607950">2.</span> <span m="609740">1/2.</span> <span m="610480">It's</span> <span m="610650">the</span> <span m="610730">probability</span> <span m="611280">that</span> <span m="611410">R</span> <span m="611730">equals</span> <span m="612080">2,</span> <span m="612520">plus the</span> <span m="612790">probability that</span> <span m="613130">R</span> <span m="613410">equals</span> <span m="613690">3.</span> <span m="614010">There's</span> <span m="614210">four</span> <span m="614470">outcomes</span> <span m="614880">there,</span> <span m="615620">each</span> <span m="615820">with</span> <span m="615940">1/8,</span> <span m="616300">so</span> <span m="616380">that's</span> <span m="616590">1/2.</span> </p>
<p><span m="621300">And</span> <span m="621610">finally,</span> <span m="622090">we</span> <span m="622240">can</span> <span m="622400">talk</span> <span m="622670">about</span> <span m="623380">the</span> <span m="623500">probability</span> <span m="624030">that</span> <span m="624280">R</span> <span m="624850">is</span> <span m="625060">in</span> <span m="625250">some</span> <span m="625540">set</span> <span m="626340">of</span> <span m="626510">values.</span> <span m="628770">So</span> <span m="628900">we</span> <span m="629030">could</span> <span m="629210">say,</span> <span m="630760">for</span> <span m="630860">example,</span> <span m="632690">for</span> <span m="633050">any</span> <span m="634210">subset</span> <span m="634670">of</span> <span m="634720">the</span> <span m="634810">real</span> <span m="635020">numbers,</span> <span m="637550">we</span> <span m="637660">define</span> <span m="638150">the</span> <span m="638240">probability</span> <span m="638830">that</span> <span m="639230">R</span> <span m="639670">attains</span> <span m="640230">a</span> <span m="640320">value</span> <span m="640890">in</span> <span m="641150">that</span> <span m="641390">set</span> <span m="643570">is</span> <span m="643720">simply</span> <span m="644100">the</span> <span m="644220">sum</span> <span m="645660">over</span> <span m="645930">all</span> <span m="646100">possible</span> <span m="646520">values</span> <span m="646940">in</span> <span m="646990">the</span> <span m="647080">set,</span> <span m="650660">that</span> <span m="650760">the</span> <span m="650890">probability</span> <span m="651500">R</span> <span m="651720">equals</span> <span m="652000">that</span> <span m="652200">value.</span> <span m="655570">So</span> <span m="655960">we</span> <span m="657140">could</span> <span m="657300">ask,</span> <span m="657710">what's</span> <span m="657850">the</span> <span m="657950">probability</span> <span m="658670">that,</span> <span m="659830">say,</span> <span m="660060">we</span> <span m="660170">take</span> <span m="660470">A</span> <span m="660850">being</span> <span m="662120">the</span> <span m="662240">set</span> <span m="662510">1,</span> <span m="662740">3.</span> <span m="664950">I</span> <span m="665080">could</span> <span m="665270">ask</span> <span m="665600">for</span> <span m="666010">the</span> <span m="666110">probability</span> <span m="666950">R</span> <span m="668120">is</span> <span m="668330">A</span> <span m="668710">when</span> <span m="669090">it's</span> <span m="669210">1,</span> <span m="669440">3,</span> <span m="669860">which</span> <span m="670070">is</span> <span m="670160">the</span> <span m="670260">same</span> <span m="670640">as</span> <span m="672400">the</span> <span m="672510">probability</span> <span m="673380">R</span> <span m="673680">is</span> <span m="673840">odd.</span> <span m="675220">Because</span> <span m="675400">Our</span> <span m="675620">could</span> <span m="675720">only</span> <span m="675920">be</span> <span m="676040">0,</span> <span m="676350">1,</span> <span m="676610">2,</span> <span m="676810">or</span> <span m="676900">3.</span> </p>
<p><span m="678210">What's</span> <span m="678420">the</span> <span m="678490">probability</span> <span m="679050">R</span> <span m="679350">is</span> <span m="679480">odd</span> <span m="681730">in</span> <span m="682190">this</span> <span m="682410">example?</span> <span m="684560">Flip</span> <span m="684820">three</span> <span m="685050">coins,</span> <span m="685550">we're asking</span> <span m="685980">the</span> <span m="686060">probability</span> <span m="686410">you</span> <span m="686760">get</span> <span m="686800">an</span> <span m="686980">odd</span> <span m="687350">number</span> <span m="687650">of</span> <span m="687730">heads--</span> <span m="688120">one</span> <span m="688510">or</span> <span m="688660">three</span> <span m="688930">heads.</span> <span m="691240">1/2.</span> <span m="692650">Again,</span> <span m="693020">you've</span> <span m="693150">got</span> <span m="693950">four</span> <span m="694450">sample</span> <span m="694940">points--</span> <span m="696820">here</span> <span m="698040">and</span> <span m="698170">there.</span> <span m="700080">So</span> <span m="700230">that's</span> <span m="700420">1/2.</span> <span m="704820">All right,</span> <span m="704990">any</span> <span m="705180">questions</span> <span m="707260">about</span> <span m="707720">the</span> <span m="707800">definition?</span> </p>
<p><span m="711210">All</span> <span m="711670">right,</span> <span m="711900">now,</span> <span m="712580">we're</span> <span m="712810">only</span> <span m="713030">going</span> <span m="713130">to</span> <span m="713210">worry</span> <span m="713510">about</span> <span m="713710">the</span> <span m="713810">case</span> <span m="714130">when</span> <span m="714290">random</span> <span m="714690">variables</span> <span m="716120">are</span> <span m="716710">discrete.</span> <span m="718100">Namely,</span> <span m="720370">there's</span> <span m="720590">a</span> <span m="720640">finite</span> <span m="721160">number</span> <span m="721810">of</span> <span m="722000">values.</span> <span m="723120">If</span> <span m="723270">you</span> <span m="723380">have</span> <span m="723560">continuous</span> <span m="724090">random</span> <span m="724360">variables,</span> <span m="724870">which</span> <span m="724940">they'll</span> <span m="725190">deal</span> <span m="725380">with in</span> <span m="725520">other</span> <span m="725760">courses,</span> <span m="726280">instead</span> <span m="726550">of</span> <span m="726610">doing</span> <span m="726850">sums</span> <span m="727360">and</span> <span m="727470">stuff</span> <span m="727660">like</span> <span m="727810">that,</span> <span m="728030">you're</span> <span m="728220">working</span> <span m="728520">with</span> <span m="728680">integrals.</span> <span m="729260">But</span> <span m="729730">for</span> <span m="729830">this</span> <span m="730010">course,</span> <span m="730300">it's</span> <span m="730450">going</span> <span m="730570">to</span> <span m="730630">be</span> <span m="731470">all</span> <span m="731760">countable</span> <span m="732230">sets,</span> <span m="732840">and</span> <span m="733010">usually</span> <span m="733600">often</span> <span m="733950">finite</span> <span m="734410">sets.</span> </p>
<p><span m="736770">Now,</span> <span m="738300">conditional</span> <span m="738840">probability</span> <span m="739500">carries</span> <span m="739920">over</span> <span m="740100">very</span> <span m="740360">nicely</span> <span m="741300">to</span> <span m="741430">random</span> <span m="741710">variables</span> <span m="742120">as</span> <span m="742220">well.</span> <span m="742690">For</span> <span m="742790">example,</span> <span m="744870">we</span> <span m="744940">could</span> <span m="745080">write</span> <span m="745360">down</span> <span m="746340">the</span> <span m="746450">probability</span> <span m="747160">that</span> <span m="747800">R</span> <span m="748320">equals</span> <span m="748810">2</span> <span m="749830">given</span> <span m="750090">that</span> <span m="750290">M</span> <span m="751120">equals</span> <span m="751550">1,</span> <span m="753530">where</span> <span m="753740">R and M</span> <span m="754020">are</span> <span m="754280">the</span> <span m="754390">same</span> <span m="754660">random</span> <span m="754910">variables</span> <span m="755290">we've</span> <span m="755420">been</span> <span m="755490">looking</span> <span m="755840">at</span> <span m="756080">when you</span> <span m="756200">flip the</span> <span m="756430">three</span> <span m="756710">coins.</span> <span m="758340">What</span> <span m="758510">is</span> <span m="758620">that</span> <span m="758810">probability?</span> <span m="760830">0,</span> <span m="761280">the</span> <span m="761380">probability</span> <span m="761890">of</span> <span m="761950">getting</span> <span m="762220">exactly</span> <span m="762720">two</span> <span m="762930">heads</span> <span m="763660">when</span> <span m="763810">they're</span> <span m="763960">all</span> <span m="764140">the</span> <span m="764230">same,</span> <span m="765650">can't</span> <span m="765870">happen.</span> <span m="766210">It's</span> <span m="766420">0.</span> <span m="766990">You can</span> <span m="767310">either</span> <span m="767610">have</span> <span m="767800">three</span> <span m="768150">heads</span> <span m="768580">or</span> <span m="769250">no</span> <span m="769470">heads.</span> <span m="770590">So</span> <span m="770760">that's</span> <span m="770970">0.</span> </p>
<p><span m="774920">The</span> <span m="775060">notion</span> <span m="775390">of</span> <span m="775470">independence</span> <span m="776450">also</span> <span m="776790">carries</span> <span m="777170">over</span> <span m="777970">to</span> <span m="778080">random</span> <span m="778360">variables,</span> <span m="778820">but</span> <span m="779040">you've</span> <span m="779160">got</span> <span m="779310">to</span> <span m="779360">be</span> <span m="779450">a</span> <span m="779500">little</span> <span m="779700">careful</span> <span m="780100">here.</span> <span m="786670">Two</span> <span m="786920">random</span> <span m="787260">variables,</span> <span m="789710">R1</span> <span m="790640">and</span> <span m="790780">R2,</span> <span m="792740">are</span> <span m="792870">said</span> <span m="793160">to</span> <span m="793220">be</span> <span m="793730">independent</span> <span m="802740">if--</span> <span m="803020">and</span> <span m="803300">this</span> <span m="803470">is</span> <span m="804120">a little</span> <span m="804330">complicated--</span> <span m="806070">for</span> <span m="806330">all</span> <span m="806720">possible</span> <span m="807360">values,</span> <span m="808940">x1</span> <span m="809385">and</span> <span m="809830">x2</span> <span m="811392">in</span> <span m="811810">the</span> <span m="811910">real</span> <span m="812120">numbers,</span> <span m="815380">the</span> <span m="815470">probability</span> <span m="817680">that</span> <span m="818330">R1</span> <span m="818970">is</span> <span m="819160">x1,</span> <span m="820290">given</span> <span m="820610">that</span> <span m="820790">R2</span> <span m="821820">is</span> <span m="821960">x2,</span> <span m="824300">is</span> <span m="824500">the</span> <span m="824610">same</span> <span m="825210">as</span> <span m="825650">the</span> <span m="825730">probability</span> <span m="826160">of</span> <span m="826240">R1</span> <span m="826580">equals</span> <span m="826830">x1</span> <span m="827320">not</span> <span m="827540">knowing</span> <span m="827790">anything</span> <span m="828490">about</span> <span m="828730">R2.</span> <span m="834060">Or</span> <span m="834680">there's</span> <span m="834870">a</span> <span m="834930">special</span> <span m="835300">case</span> <span m="835620">when</span> <span m="835740">this</span> <span m="835910">can't</span> <span m="836180">happen.</span> <span m="837610">Namely,</span> <span m="838480">the</span> <span m="838580">probability</span> <span m="839270">that</span> <span m="839490">R2</span> <span m="840050">equals</span> <span m="840400">x2</span> <span m="841650">is</span> <span m="841750">0.</span> </p>
<p><span m="845020">So,</span> <span m="845630">for</span> <span m="845900">two</span> <span m="846160">random</span> <span m="846510">variables</span> <span m="846970">to</span> <span m="847050">be</span> <span m="847210">independent,</span> <span m="848830">it</span> <span m="849050">needs</span> <span m="849360">to</span> <span m="849460">be</span> <span m="849600">the</span> <span m="849720">case</span> <span m="850320">that,</span> <span m="851160">no</span> <span m="851230">matter</span> <span m="851490">what</span> <span m="851810">happens</span> <span m="852300">out</span> <span m="852470">here</span> <span m="852720">with</span> <span m="852870">R2,</span> <span m="853670">and</span> <span m="853780">whatever</span> <span m="854140">you</span> <span m="854250">know</span> <span m="854480">about</span> <span m="854830">it,</span> <span m="855910">it</span> <span m="856070">can't</span> <span m="856470">influence</span> <span m="857030">the</span> <span m="857130">probability</span> <span m="858850">for</span> <span m="859040">R1</span> <span m="859740">to</span> <span m="859820">equal</span> <span m="860210">anything.</span> <span m="862030">So</span> <span m="862240">no</span> <span m="862710">value</span> <span m="863250">of</span> <span m="863360">R2</span> <span m="863960">influences</span> <span m="864610">any</span> <span m="864940">value</span> <span m="865370">of</span> <span m="865490">R1.</span> <span m="867790">So</span> <span m="867990">it's</span> <span m="868130">the</span> <span m="868510">strongest</span> <span m="869320">possible</span> <span m="869740">thing</span> <span m="870140">with</span> <span m="870300">independence</span> <span m="871180">for</span> <span m="871400">it to</span> <span m="871500">apply</span> <span m="872010">to</span> <span m="872160">random</span> <span m="872430">variables.</span> </p>
<p><span m="873946">And</span> <span m="874390">there's</span> <span m="874530">an</span> <span m="874610">equivalent</span> <span m="875240">definition,</span> <span m="876570">and</span> <span m="876720">we're</span> <span m="876840">going</span> <span m="876940">to</span> <span m="876990">use</span> <span m="877340">these</span> <span m="877570">interchangeably,</span> <span m="878700">just</span> <span m="879040">as</span> <span m="879120">there</span> <span m="879260">were</span> <span m="879510">two</span> <span m="879680">definitions</span> <span m="880380">for</span> <span m="880550">independence</span> <span m="881660">of</span> <span m="881830">events,</span> <span m="883010">there's</span> <span m="883220">the</span> <span m="883280">product</span> <span m="883810">for them.</span> <span m="885840">So</span> <span m="885970">the</span> <span m="886070">equivalent</span> <span m="886460">definitions</span> <span m="887270">says</span> <span m="888890">for</span> <span m="889120">all</span> <span m="889420">x1</span> <span m="889880">and</span> <span m="890340">x2</span> <span m="891810">in</span> <span m="891920">the</span> <span m="892010">real</span> <span m="892210">numbers,</span> <span m="894750">the</span> <span m="894860">probability</span> <span m="895500">that</span> <span m="895680">R1</span> <span m="896870">equals</span> <span m="897370">x1</span> <span m="898160">and</span> <span m="899270">R2</span> <span m="899840">equals</span> <span m="900250">x2</span> <span m="902070">is</span> <span m="902240">simply</span> <span m="902550">the</span> <span m="902660">product</span> <span m="903350">of</span> <span m="903420">those</span> <span m="903700">probabilities.</span> <span m="905680">Probability</span> <span m="906100">R1</span> <span m="906440">equals</span> <span m="906680">x1</span> <span m="907630">times</span> <span m="908790">the</span> <span m="908880">probability</span> <span m="910510">R2</span> <span m="911570">equals</span> <span m="911950">x2.</span> </p>
<p><span m="915180">And</span> <span m="915370">you</span> <span m="915500">don't have</span> <span m="915680">to</span> <span m="915760">worry</span> <span m="916050">about</span> <span m="916280">that</span> <span m="916490">case</span> <span m="916820">where</span> <span m="916950">R2</span> <span m="917523">being</span> <span m="917876">x2</span> <span m="918230">is</span> <span m="918340">0.</span> <span m="919980">So</span> <span m="920140">we</span> <span m="920290">can</span> <span m="920430">use</span> <span m="920620">both</span> <span m="920970">of</span> <span m="921050">those</span> <span m="921980">as</span> <span m="922220">definitions</span> <span m="923440">of</span> <span m="923510">independence</span> <span m="924490">of</span> <span m="924630">random</span> <span m="924950">variables.</span> </p>
<p><span m="927550">All</span> <span m="927600">right,</span> <span m="927840">so</span> <span m="928490">let's</span> <span m="928700">see</span> <span m="928840">an</span> <span m="928910">example.</span> <span m="929430">We've</span> <span m="929580">got</span> <span m="929930">R and</span> <span m="930160">M.</span> <span m="931000">We've</span> <span m="931160">talked</span> <span m="931420">about</span> <span m="931670">two</span> <span m="931840">random</span> <span m="932110">variables.</span> <span m="933500">Are</span> <span m="933640">they</span> <span m="934060">independent</span> <span m="935630">random</span> <span m="936000">variables?</span> <span m="940610">M</span> <span m="941280">is</span> <span m="941450">the</span> <span m="941600">indicated</span> <span m="942120">random</span> <span m="942420">variable</span> <span m="942790">for</span> <span m="942930">all</span> <span m="943140">the</span> <span m="943240">coins</span> <span m="943560">matching.</span> <span m="944630">R</span> <span m="945070">is</span> <span m="945180">the</span> <span m="945290">random</span> <span m="945600">variable</span> <span m="945970">that</span> <span m="946060">tells</span> <span m="946400">us</span> <span m="946520">how</span> <span m="946680">many</span> <span m="946900">heads</span> <span m="947180">there</span> <span m="947330">were in</span> <span m="947600">three</span> <span m="947790">coins.</span> <span m="949210">Are</span> <span m="949370">they</span> <span m="949510">independent?</span> </p>
<p><span m="950790">AUDIENCE: No.</span> </p>
<p><span m="951510">PROFESSOR: No,</span> <span m="952350">why</span> <span m="952580">not?</span> </p>
<p><span m="954024">AUDIENCE: Because</span> <span m="954491">information</span> <span m="954958">with what</span> <span m="955425">one is</span> <span m="956359">narrows</span> <span m="956826">down</span> <span m="957293">[INAUDIBLE].</span> </p>
<p><span m="958230">PROFESSOR: Yes,</span> <span m="958680">that's</span> <span m="958930">true.</span> <span m="959100">Information</span> <span m="959650">about</span> <span m="959840">what</span> <span m="959980">one</span> <span m="960290">value</span> <span m="960740">is</span> <span m="961560">influences</span> <span m="962170">the</span> <span m="962260">probability</span> <span m="963330">that</span> <span m="963490">the</span> <span m="963630">other</span> <span m="963830">guy</span> <span m="964210">has</span> <span m="964430">a</span> <span m="964500">certain</span> <span m="964800">value.</span> <span m="965240">In</span> <span m="965390">particular,</span> <span m="966800">we</span> <span m="966940">can</span> <span m="967090">find</span> <span m="967560">a</span> <span m="967640">case</span> <span m="968160">of</span> <span m="968260">x1</span> <span m="968660">and</span> <span m="968830">x2</span> <span m="969030">where</span> <span m="969120">this</span> <span m="969340">fails.</span> <span m="971020">One</span> <span m="971270">case</span> <span m="971590">would</span> <span m="971720">be</span> <span m="972070">the</span> <span m="972180">probability</span> <span m="972910">of</span> <span m="973100">what we've</span> <span m="973290">already</span> <span m="973500">done,</span> <span m="974510">that</span> <span m="974800">R</span> <span m="975220">equals</span> <span m="975950">2</span> <span m="977510">and</span> <span m="978380">M</span> <span m="978820">equals</span> <span m="979260">1.</span> <span m="981036">What</span> <span m="981480">was</span> <span m="981600">this</span> <span m="981760">probability?</span> <span m="983930">0.</span> <span m="984650">If</span> <span m="985030">everything</span> <span m="985460">matches,</span> <span m="985910">you</span> <span m="986030">can't</span> <span m="986250">have</span> <span m="986390">two</span> <span m="986540">heads.</span> <span m="986780">That's</span> <span m="986980">0.</span> <span m="988110">That</span> <span m="988300">does</span> <span m="988560">not</span> <span m="988880">equal</span> <span m="989290">the</span> <span m="989390">probability</span> <span m="991220">of</span> <span m="991380">two</span> <span m="991630">heads</span> <span m="992460">times</span> <span m="993450">the</span> <span m="993530">probability</span> <span m="994810">they</span> <span m="994950">all</span> <span m="995120">match.</span> <span m="996580">Because</span> <span m="996820">this</span> <span m="997970">probability,</span> <span m="998370">two</span> <span m="998560">heads</span> <span m="998850">is</span> <span m="998970">3/8,</span> <span m="1000550">probability</span> <span m="1000930">they</span> <span m="1001310">all</span> <span m="1001440">match is</span> <span m="1001860">1/4.</span> <span m="1003045">And</span> <span m="1003520">3/8</span> <span m="1003810">times</span> <span m="1004180">a</span> <span m="1004240">1/4</span> <span m="1004650">is</span> <span m="1004870">not</span> <span m="1005250">0.</span> <span m="1007510">So</span> <span m="1008140">R and</span> <span m="1008410">M are</span> <span m="1008830">not</span> <span m="1009390">independent.</span> </p>
<p><span m="1017470">OK,</span> <span m="1017710">now</span> <span m="1017890">in</span> <span m="1017990">general</span> <span m="1018450">to</span> <span m="1018530">show</span> <span m="1019520">two</span> <span m="1019710">random</span> <span m="1020040">variables</span> <span m="1020560">are</span> <span m="1020730">not</span> <span m="1021210">independent,</span> <span m="1022275">all</span> <span m="1022700">you</span> <span m="1022830">gotta</span> <span m="1023070">do</span> <span m="1023270">is</span> <span m="1023420">find</span> <span m="1023820">one</span> <span m="1024900">pair</span> <span m="1025230">of</span> <span m="1026290">[? out ?]</span> <span m="1026579">values</span> <span m="1027200">for</span> <span m="1027349">which</span> <span m="1027950">there's</span> <span m="1028180">dependence.</span> <span m="1029700">To</span> <span m="1029849">show</span> <span m="1030079">their</span> <span m="1030270">independent,</span> <span m="1031930">you've</span> <span m="1032060">got</span> <span m="1032230">to</span> <span m="1032300">deal</span> <span m="1032540">with</span> <span m="1032660">all</span> <span m="1033369">possible</span> <span m="1033910">pairs</span> <span m="1034230">of</span> <span m="1034319">values.</span> <span m="1035390">So</span> <span m="1035490">it's</span> <span m="1035589">harder</span> <span m="1036050">show</span> <span m="1036390">independence</span> <span m="1037020">in</span> <span m="1037099">general.</span> </p>
<p><span m="1040720">All</span> <span m="1040869">right,</span> <span m="1041280">let's</span> <span m="1041510">do</span> <span m="1041700">another</span> <span m="1041960">example,</span> <span m="1044588">get</span> <span m="1045060">a</span> <span m="1045109">little</span> <span m="1045369">practice</span> <span m="1045774">with</span> <span m="1046180">this.</span> <span m="1058180">Say</span> <span m="1058440">we</span> <span m="1058620">have</span> <span m="1058910">two</span> <span m="1059370">fair</span> <span m="1060260">independent</span> <span m="1062050">six-sided</span> <span m="1062810">die,</span> <span m="1070130">normal</span> <span m="1070520">dice.</span> <span m="1071730">And</span> <span m="1071950">the</span> <span m="1072110">outcome</span> <span m="1072390">of</span> <span m="1072440">the</span> <span m="1072510">first</span> <span m="1072870">one</span> <span m="1072990">is</span> <span m="1073120">D1,</span> <span m="1074590">and</span> <span m="1074720">the</span> <span m="1074830">outcome</span> <span m="1075120">of</span> <span m="1075160">the</span> <span m="1075260">second</span> <span m="1075620">roll</span> <span m="1075910">is</span> <span m="1076040">D2.</span> <span m="1078710">And</span> <span m="1079520">by</span> <span m="1079650">independent,</span> <span m="1080340">I</span> <span m="1080450">mean</span> <span m="1080740">here</span> <span m="1081030">that</span> <span m="1081740">knowing</span> <span m="1082690">any</span> <span m="1083550">information</span> <span m="1084170">about</span> <span m="1084390">what</span> <span m="1084540">the</span> <span m="1084630">second</span> <span m="1084990">die</span> <span m="1085200">is</span> <span m="1085620">does</span> <span m="1085740">not</span> <span m="1086040">give</span> <span m="1086180">you</span> <span m="1086340">any</span> <span m="1087700">change in</span> <span m="1088310">your</span> <span m="1088410">probability</span> <span m="1089020">that</span> <span m="1089160">the</span> <span m="1089230">first</span> <span m="1089550">die</span> <span m="1089780">has</span> <span m="1089930">any</span> <span m="1090100">value.</span> </p>
<p><span m="1092690">And</span> <span m="1092800">now</span> <span m="1092980">we</span> <span m="1093080">define</span> <span m="1093480">another</span> <span m="1094650">value,</span> <span m="1095060">S,</span> <span m="1095500">to</span> <span m="1095620">be</span> <span m="1096464">D1</span> <span m="1097308">plus</span> <span m="1097730">D2,</span> <span m="1098360">the</span> <span m="1098460">sum</span> <span m="1099190">of</span> <span m="1099420">the</span> <span m="1099520">dice.</span> <span m="1102490">Is</span> <span m="1102750">S</span> <span m="1103120">a</span> <span m="1103190">random</span> <span m="1103490">variable?</span> <span m="1107920">Yeah,</span> <span m="1108930">for</span> <span m="1109130">any</span> <span m="1109390">outcome--</span> <span m="1110040">an</span> <span m="1110200">outcome</span> <span m="1110520">being</span> <span m="1111250">a</span> <span m="1111560">pair of</span> <span m="1111875">dice</span> <span m="1112190">values--</span> <span m="1113290">it</span> <span m="1113480">maps</span> <span m="1113840">that</span> <span m="1114030">outcome</span> <span m="1114490">to</span> <span m="1114710">a</span> <span m="1114920">real</span> <span m="1115160">number,</span> <span m="1116026">a number</span> <span m="1116460">between</span> <span m="1117190">2</span> <span m="1117590">and</span> <span m="1118010">12,</span> <span m="1120170">OK?</span> </p>
<p><span m="1121700">Let's</span> <span m="1122070">do</span> <span m="1122390">another</span> <span m="1122750">one.</span> <span m="1123560">Let's</span> <span m="1123750">let</span> <span m="1125670">T</span> <span m="1128095">be</span> <span m="1129310">1</span> <span m="1130860">if</span> <span m="1131330">S</span> <span m="1131625">is</span> <span m="1131920">7,</span> <span m="1132800">namely</span> <span m="1132920">the</span> <span m="1133160">sum</span> <span m="1133460">of</span> <span m="1133500">the</span> <span m="1133580">dice</span> <span m="1133870">is</span> <span m="1133920">7,</span> <span m="1134330">and</span> <span m="1134470">0</span> <span m="1134860">otherwise.</span> <span m="1139860">T</span> <span m="1140290">is</span> <span m="1140440">also</span> <span m="1140780">a</span> <span m="1140860">random</span> <span m="1141180">variable.</span> <span m="1142340">In</span> <span m="1142440">fact,</span> <span m="1142570">what</span> <span m="1142740">kind</span> <span m="1143080">is</span> <span m="1143220">it?</span> <span m="1145030">Yeah,</span> <span m="1145340">indicator,</span> <span m="1146200">characteristic,</span> <span m="1147550">whatever.</span> <span m="1148550">It's</span> <span m="1148690">telling</span> <span m="1149040">you</span> <span m="1149190">if</span> <span m="1149280">the</span> <span m="1149570">sum</span> <span m="1149860">of</span> <span m="1149900">the</span> <span m="1149990">dice</span> <span m="1150320">is</span> <span m="1150430">7</span> <span m="1150950">or</span> <span m="1151130">not.</span> </p>
<p><span m="1153830">All</span> <span m="1154040">right,</span> <span m="1154350">now,</span> <span m="1155590">there's</span> <span m="1155790">four</span> <span m="1156150">random</span> <span m="1156470">variables</span> <span m="1156950">here.</span> <span m="1158300">Each</span> <span m="1158590">die,</span> <span m="1160020">which</span> <span m="1160230">we</span> <span m="1160390">already</span> <span m="1160690">have</span> <span m="1160830">told</span> <span m="1161160">you,</span> <span m="1161440">we're</span> <span m="1161580">assuming</span> <span m="1162020">are</span> <span m="1162110">independent.</span> <span m="1163370">The sum</span> <span m="1164630">and</span> <span m="1164770">the</span> <span m="1164970">indicator</span> <span m="1165820">if</span> <span m="1165870">the</span> <span m="1165960">sum</span> <span m="1166270">is</span> <span m="1166330">7.</span> <span m="1170130">All</span> <span m="1170260">right,</span> <span m="1171740">what</span> <span m="1171880">about</span> <span m="1172110">these</span> <span m="1172320">two</span> <span m="1172460">values?</span> <span m="1174990">Are</span> <span m="1175490">D1,</span> <span m="1175940">the</span> <span m="1176020">first</span> <span m="1176360">die,</span> <span m="1176850">and</span> <span m="1176980">the</span> <span m="1177060">sum</span> <span m="1177990">independent?</span> </p>
<p><span m="1181603">AUDIENCE: No?</span> <span m="1183998">No?</span> </p>
<p><span m="1185440">PROFESSOR: No,</span> <span m="1186250">intuitively</span> <span m="1186740">they're</span> <span m="1186930">not.</span> <span m="1187850">Because</span> <span m="1189870">if</span> <span m="1190080">I</span> <span m="1190180">know</span> <span m="1190480">something</span> <span m="1190920">about</span> <span m="1191180">the</span> <span m="1191270">first</span> <span m="1191620">die,</span> <span m="1192020">I</span> <span m="1192110">probably</span> <span m="1193230">know</span> <span m="1193410">something</span> <span m="1193730">about</span> <span m="1194060">the</span> <span m="1194140">sum.</span> <span m="1195710">Now</span> <span m="1195910">to</span> <span m="1196010">really</span> <span m="1196330">nail</span> <span m="1196670">that</span> <span m="1196940">down,</span> <span m="1198990">you've</span> <span m="1199110">got</span> <span m="1199320">to</span> <span m="1200260">give</span> <span m="1200410">me</span> <span m="1200590">a</span> <span m="1200660">value</span> <span m="1201350">for</span> <span m="1201560">this,</span> <span m="1202370">which</span> <span m="1202670">influences</span> <span m="1203220">the</span> <span m="1203310">probability</span> <span m="1203880">that</span> <span m="1204120">obtains</span> <span m="1204540">some</span> <span m="1204850">value,</span> <span m="1207290">right?</span> <span m="1207560">That's</span> <span m="1207750">how</span> <span m="1207880">you</span> <span m="1208080">would</span> <span m="1208320">convince</span> <span m="1208860">me</span> <span m="1209020">that,</span> <span m="1209150">in</span> <span m="1209250">fact,</span> <span m="1209590">they're</span> <span m="1209710">dependent.</span> </p>
<p><span m="1213600">Can</span> <span m="1214090">anybody</span> <span m="1214430">give</span> <span m="1214610">me</span> <span m="1214710">a</span> <span m="1214760">value</span> <span m="1215170">for</span> <span m="1215310">this</span> <span m="1215870">that</span> <span m="1216840">changes</span> <span m="1217320">the</span> <span m="1217400">probability</span> <span m="1217910">this</span> <span m="1218120">equals</span> <span m="1218420">something?</span> </p>
<p><span m="1223320">AUDIENCE: Guess?</span> <span m="1224250">If it's</span> <span m="1224690">like</span> <span m="1225130">1,</span> <span m="1225570">then</span> <span m="1226010">[INAUDIBLE]</span> <span m="1226450">has to be</span> <span m="1226890">6.</span> </p>
<p><span m="1227330">PROFESSOR: If</span> <span m="1228050">this</span> <span m="1228230">is</span> <span m="1228460">1,</span> <span m="1231220">what</span> <span m="1231460">value</span> <span m="1231770">of</span> <span m="1231860">this</span> <span m="1232170">could</span> <span m="1232310">be</span> <span m="1232480">influenced?</span> </p>
<p><span m="1235028">AUDIENCE: You know that</span> <span m="1236012">in</span> <span m="1236504">order</span> <span m="1236996">for--</span> <span m="1239230">the only</span> <span m="1239500">set of</span> <span m="1239700">values</span> <span m="1239930">S</span> <span m="1240260">can</span> <span m="1240640">take</span> <span m="1241020">now,</span> <span m="1241400">it  can</span> <span m="1241725">only</span> <span m="1242050">be from</span> <span m="1242776">1</span> <span m="1243140">to--</span> <span m="1243530">from 2</span> <span m="1243920">to  7.</span> </p>
<p><span m="1244382">PROFESSOR: From 2 to</span> <span m="1244844">7.</span> <span m="1245306">In</span> <span m="1245770">particular</span> <span m="1246190">now,</span> <span m="1246610">this</span> <span m="1247110">can't</span> <span m="1247610">be</span> <span m="1248015">12.</span> <span m="1249230">So</span> <span m="1249300">what</span> <span m="1249410">I</span> <span m="1249490">could</span> <span m="1249660">do</span> <span m="1249850">is</span> <span m="1249980">say</span> <span m="1250310">the</span> <span m="1250410">probability</span> <span m="1251070">that</span> <span m="1251240">S</span> <span m="1251570">equals</span> <span m="1252020">12</span> <span m="1253780">and</span> <span m="1255510">the</span> <span m="1255600">first</span> <span m="1255940">die</span> <span m="1256270">was</span> <span m="1256480">1,</span> <span m="1257180">what's</span> <span m="1257410">that</span> <span m="1257580">probability?</span> </p>
<p><span m="1258800">AUDIENCE: 0.</span> </p>
<p><span m="1259280">PROFESSOR: 0.</span> <span m="1261300">And</span> <span m="1261480">that</span> <span m="1261640">does</span> <span m="1261840">not</span> <span m="1262230">equal</span> <span m="1263190">the</span> <span m="1263290">probability</span> <span m="1263980">S</span> <span m="1264240">is</span> <span m="1264470">12</span> <span m="1265570">times</span> <span m="1266550">the</span> <span m="1266650">probability</span> <span m="1267180">the</span> <span m="1267270">first</span> <span m="1267580">die</span> <span m="1267900">was</span> <span m="1268120">1.</span> <span m="1271240">Or</span> <span m="1271430">we</span> <span m="1271540">could</span> <span m="1271670">plug</span> <span m="1271920">it</span> <span m="1271970">into</span> <span m="1272150">the</span> <span m="1272270">other</span> <span m="1272440">definition.</span> <span m="1273690">The</span> <span m="1273790">probability</span> <span m="1274250">that</span> <span m="1274330">S</span> <span m="1274580">equals</span> <span m="1274860">12</span> <span m="1275210">given</span> <span m="1275700">the</span> <span m="1275780">first</span> <span m="1276100">die is</span> <span m="1276480">1</span> <span m="1278290">is</span> <span m="1278460">not</span> <span m="1278690">equal</span> <span m="1278940">to</span> <span m="1278990">the</span> <span m="1279080">probability</span> <span m="1279590">S</span> <span m="1279780">equals</span> <span m="1280050">12.</span> <span m="1281440">And,</span> <span m="1281680">in</span> <span m="1281760">fact,</span> <span m="1282400">probability</span> <span m="1282840">the</span> <span m="1282920">first</span> <span m="1283220">die is</span> <span m="1283480">1</span> <span m="1283700">is</span> <span m="1283830">not</span> <span m="1284030">0.</span> <span m="1284510">So</span> <span m="1284670">either</span> <span m="1284930">definition,</span> <span m="1285420">of</span> <span m="1285490">course,</span> <span m="1286340">works</span> <span m="1286750">to</span> <span m="1286850">show</span> <span m="1287220">that</span> <span m="1287540">they</span> <span m="1287780">are</span> <span m="1288140">dependent.</span> <span m="1289630">So</span> <span m="1289820">S</span> <span m="1290090">and</span> <span m="1290240">D1</span> <span m="1290575">are</span> <span m="1290910">dependent.</span> <span m="1295800">All</span> <span m="1296100">you</span> <span m="1296180">need</span> <span m="1296340">is</span> <span m="1296450">one</span> <span m="1297830">possible</span> <span m="1298230">pair</span> <span m="1298430">of</span> <span m="1298490">values</span> <span m="1298940">to</span> <span m="1299070">show</span> <span m="1299270">dependence.</span> <span m="1300360">That's</span> <span m="1300600">usually</span> <span m="1301040">easy</span> <span m="1301280">to</span> <span m="1301360">do.</span> </p>
<p><span m="1306350">All</span> <span m="1306460">right,</span> <span m="1306750">what</span> <span m="1307000">about</span> <span m="1309220">D1,</span> <span m="1309910">the</span> <span m="1310000">first</span> <span m="1310500">die,</span> <span m="1311020">and</span> <span m="1311380">T,</span> <span m="1312700">the</span> <span m="1312860">indicator</span> <span m="1314330">for</span> <span m="1315000">getting</span> <span m="1315290">a</span> <span m="1315360">7?</span> <span m="1319150">Are</span> <span m="1319405">D1</span> <span m="1320160">and</span> <span m="1320580">T</span> <span m="1321580">independent</span> <span m="1322500">random</span> <span m="1322910">variables?</span> <span m="1325815">All right,</span> <span m="1326240">somebody's</span> <span m="1326650">shaking</span> <span m="1326730">their</span> <span m="1326860">head</span> <span m="1327100">no.</span> <span m="1328860">They</span> <span m="1329090">seem</span> <span m="1329400">to</span> <span m="1329460">be</span> <span m="1329570">dependent</span> <span m="1330100">because</span> <span m="1331000">knowing</span> <span m="1331410">the</span> <span m="1331510">first</span> <span m="1331940">die</span> <span m="1332430">seems</span> <span m="1332880">like</span> <span m="1333100">it</span> <span m="1333180">should</span> <span m="1333360">tell</span> <span m="1333560">you</span> <span m="1333710">something</span> <span m="1334470">about</span> <span m="1334710">the</span> <span m="1334810">probability</span> <span m="1335660">of</span> <span m="1335780">getting</span> <span m="1336390">a</span> <span m="1336470">7</span> <span m="1337000">for</span> <span m="1337100">the</span> <span m="1337210">sum.</span> <span m="1339930">That's</span> <span m="1340140">a</span> <span m="1340200">good</span> <span m="1340560">first</span> <span m="1340890">intuition.</span> <span m="1342140">In</span> <span m="1342220">fact,</span> <span m="1342460">you</span> <span m="1342680">always</span> <span m="1343090">want</span> <span m="1343210">to</span> <span m="1343280">assume</span> <span m="1343770">things</span> <span m="1344000">are</span> <span m="1344040">dependent</span> <span m="1344520">unless</span> <span m="1344800">you</span> <span m="1345820">convince</span> <span m="1346200">yourself</span> <span m="1346600">otherwise.</span> <span m="1347070">It's</span> <span m="1347200">always</span> <span m="1347450">a</span> <span m="1347520">good</span> <span m="1348760">rule.</span> </p>
<p><span m="1349770">Now,</span> <span m="1349800">in</span> <span m="1349930">fact,</span> <span m="1350250">in</span> <span m="1350320">this</span> <span m="1350510">case,</span> <span m="1351640">it</span> <span m="1351780">is</span> <span m="1352000">independent.</span> <span m="1354200">The</span> <span m="1354220">probability</span> <span m="1354880">of</span> <span m="1354960">getting</span> <span m="1355240">a</span> <span m="1355290">7</span> <span m="1356400">is</span> <span m="1356620">not,</span> <span m="1357110">it</span> <span m="1357210">turns</span> <span m="1357580">out,</span> <span m="1357910">influenced</span> <span m="1358780">by</span> <span m="1358950">the</span> <span m="1359050">first</span> <span m="1359360">die.</span> <span m="1361540">Anything</span> <span m="1361960">else</span> <span m="1362260">would</span> <span m="1362500">be,</span> <span m="1363480">but</span> <span m="1363640">7</span> <span m="1364040">is</span> <span m="1364170">not.</span> <span m="1364380">Let's</span> <span m="1364590">see</span> <span m="1364770">why</span> <span m="1364920">that</span> <span m="1365120">is.</span> <span m="1366040">And</span> <span m="1366210">to</span> <span m="1366290">do</span> <span m="1366490">that,</span> <span m="1366670">we'd</span> <span m="1366770">actually</span> <span m="1367060">have</span> <span m="1367160">to</span> <span m="1367270">check,</span> <span m="1367660">I</span> <span m="1367710">think,</span> <span m="1368020">12</span> <span m="1368510">cases,</span> <span m="1369610">all</span> <span m="1369900">possible</span> <span m="1370300">values</span> <span m="1370650">for</span> <span m="1370770">D1</span> <span m="1371180">and</span> <span m="1371320">T,</span> <span m="1371700">but</span> <span m="1371820">you'll</span> <span m="1371990">get</span> <span m="1372170">the</span> <span m="1372400">idea</span> <span m="1372740">pretty</span> <span m="1372970">quick.</span> </p>
<p><span m="1374710">The</span> <span m="1374820">probability</span> <span m="1375390">that</span> <span m="1375570">T</span> <span m="1375980">equals</span> <span m="1376290">1,</span> <span m="1378020">namely</span> <span m="1378350">you got</span> <span m="1378470">a</span> <span m="1378520">7,</span> <span m="1379620">given</span> <span m="1379870">that</span> <span m="1379970">D1</span> <span m="1380410">equals</span> <span m="1380680">1.</span> <span m="1380970">What's</span> <span m="1381190">that?</span> <span m="1383180">The</span> <span m="1383280">probability</span> <span m="1383950">of</span> <span m="1384050">getting</span> <span m="1384340">7</span> <span m="1384950">for</span> <span m="1385050">the</span> <span m="1385160">sum</span> <span m="1386190">given</span> <span m="1386410">your</span> <span m="1386510">first</span> <span m="1386820">die is</span> <span m="1387110">a</span> <span m="1387180">1,</span> <span m="1387480">what's</span> <span m="1387680">that</span> <span m="1387820">probability?</span> </p>
<p><span m="1388760">AUDIENCE: 1/6?</span> </p>
<p><span m="1389540">PROFESSOR: 1/6,</span> <span m="1390220">because</span> <span m="1390500">the second</span> <span m="1390910">one</span> <span m="1391060">better</span> <span m="1391310">be</span> <span m="1391570">a</span> <span m="1391640">6.</span> <span m="1393730">All</span> <span m="1393870">right,</span> <span m="1394120">and</span> <span m="1395710">what's</span> <span m="1395960">the</span> <span m="1396070">probability</span> <span m="1396350">T</span> <span m="1396630">equals</span> <span m="1397270">1?</span> <span m="1399500">What's</span> <span m="1399680">the</span> <span m="1399760">probability</span> <span m="1400200">of</span> <span m="1400290">rolling</span> <span m="1401080">a</span> <span m="1401150">7?</span> <span m="1404810">Yeah?</span> </p>
<p><span m="1406096">AUDIENCE: 6/36.</span> </p>
<p><span m="1407240">PROFESSOR: 6/36.</span> <span m="1408670">There's</span> <span m="1408780">six</span> <span m="1409160">ways</span> <span m="1409410">to</span> <span m="1409510">do</span> <span m="1409700">it.</span> <span m="1410160">1</span> <span m="1410380">and</span> <span m="1410480">6,</span> <span m="1411150">2</span> <span m="1411380">and</span> <span m="1411510">5,</span> <span m="1412180">3</span> <span m="1412610">and</span> <span m="1413220">4,</span> <span m="1413660">4</span> <span m="1413920">and</span> <span m="1414040">3,</span> <span m="1414700">5</span> <span m="1415020">and 2, and</span> <span m="1415210">6</span> <span m="1415440">and</span> <span m="1415540">1,</span> <span m="1415880">out</span> <span m="1416240">of</span> <span m="1416400">36</span> <span m="1417280">possibilities</span> <span m="1418070">equally</span> <span m="1418340">likely.</span> <span m="1419370">So</span> <span m="1419410">that</span> <span m="1419580">worked</span> <span m="1420080">here.</span> <span m="1422670">Now</span> <span m="1422930">probability</span> <span m="1423620">of</span> <span m="1423690">getting</span> <span m="1424950">a</span> <span m="1425060">7</span> <span m="1425520">given</span> <span m="1425730">the</span> <span m="1425810">second</span> <span m="1426240">die--</span> <span m="1426860">sorry--</span> <span m="1426970">the</span> <span m="1427060">first</span> <span m="1427450">die is a</span> <span m="1427845">2.</span> <span m="1429620">What's</span> <span m="1429820">that?</span> <span m="1433740">Probability</span> <span m="1434190">of</span> <span m="1434250">getting</span> <span m="1434460">a</span> <span m="1434510">7</span> <span m="1435080">given</span> <span m="1435300">that</span> <span m="1435340">your</span> <span m="1435450">first</span> <span m="1435800">die is</span> <span m="1436000">a</span> <span m="1436080">2.</span> <span m="1438310">1/6,</span> <span m="1439800">because</span> <span m="1440060">you've</span> <span m="1440190">got</span> <span m="1440320">to</span> <span m="1440360">have</span> <span m="1440520">the</span> <span m="1440600">second</span> <span m="1441010">die be a</span> <span m="1441310">5,</span> <span m="1442720">and</span> <span m="1442800">that</span> <span m="1442930">happens</span> <span m="1443270">1/6.</span> <span m="1445670">And</span> <span m="1445840">that</span> <span m="1445990">equals</span> <span m="1446300">the</span> <span m="1446380">right</span> <span m="1446600">thing.</span> </p>
<p><span m="1448740">In</span> <span m="1448770">fact,</span> <span m="1449910">I</span> <span m="1450010">can</span> <span m="1450070">keep</span> <span m="1450290">on</span> <span m="1450460">going</span> <span m="1450830">here.</span> <span m="1451800">Probability</span> <span m="1452360">T</span> <span m="1452630">equals</span> <span m="1452990">1,</span> <span m="1453836">given</span> <span m="1454260">D1</span> <span m="1454730">is</span> <span m="1454970">anything,</span> <span m="1456180">6.</span> <span m="1458560">Well,</span> <span m="1458750">for</span> <span m="1458910">every</span> <span m="1459220">value</span> <span m="1459560">of</span> <span m="1459640">D1,</span> <span m="1460045">there's</span> <span m="1460450">exactly</span> <span m="1460990">one</span> <span m="1461740">value</span> <span m="1462140">of</span> <span m="1462260">D2</span> <span m="1463250">that</span> <span m="1463410">adds</span> <span m="1463790">to</span> <span m="1463920">a</span> <span m="1463990">7.</span> <span m="1465690">So</span> <span m="1465920">it's</span> <span m="1466020">still</span> <span m="1466220">1/6,</span> <span m="1468790">which</span> <span m="1469030">is</span> <span m="1469150">the</span> <span m="1469240">probability</span> <span m="1469850">of</span> <span m="1470830">getting</span> <span m="1471060">the</span> <span m="1471150">7 in</span> <span m="1471460">the</span> <span m="1471510">first</span> <span m="1471780">place.</span> <span m="1473770">Now</span> <span m="1473850">we've</span> <span m="1473980">also</span> <span m="1474300">got</span> <span m="1474440">to</span> <span m="1474520">check</span> <span m="1475660">the</span> <span m="1475740">probability</span> <span m="1476430">T</span> <span m="1476815">equals</span> <span m="1477200">0,</span> <span m="1478700">given</span> <span m="1479030">all</span> <span m="1479340">these</span> <span m="1479620">cases</span> <span m="1480030">for</span> <span m="1480140">D1.</span> <span m="1480910">D1</span> <span m="1481295">equals</span> <span m="1481680">1.</span> </p>
<p><span m="1483340">What's</span> <span m="1483590">this?</span> <span m="1483850">The</span> <span m="1483950">probability</span> <span m="1484500">of</span> <span m="1484610">not</span> <span m="1484980">getting</span> <span m="1485220">a</span> <span m="1485280">7</span> <span m="1485700">given</span> <span m="1485930">the</span> <span m="1486010">first</span> <span m="1486280">die</span> <span m="1486470">is</span> <span m="1486600">a</span> <span m="1486670">1.</span> <span m="1486970">What's</span> <span m="1487170">that?</span> </p>
<p><span m="1489875">AUDIENCE: 5/6.</span> </p>
<p><span m="1490780">PROFESSOR: 5/6,</span> <span m="1491710">it's</span> <span m="1491860">the</span> <span m="1491990">opposite</span> <span m="1492390">of</span> <span m="1492480">this</span> <span m="1492690">case.</span> <span m="1493820">They</span> <span m="1493980">pair</span> <span m="1494310">up.</span> <span m="1494460">That's</span> <span m="1494670">got</span> <span m="1494790">to</span> <span m="1494840">be</span> <span m="1494950">5/6,</span> <span m="1497660">and</span> <span m="1497860">that</span> <span m="1498050">is,</span> <span m="1498220">of</span> <span m="1498300">course,</span> <span m="1499300">the</span> <span m="1499390">probability</span> <span m="1500600">of</span> <span m="1500720">T</span> <span m="1500960">equals</span> <span m="1501240">0.</span> <span m="1502960">And</span> <span m="1503290">the</span> <span m="1503360">same</span> <span m="1503720">is</span> <span m="1503840">true</span> <span m="1504280">for</span> <span m="1504580">all</span> <span m="1505000">the</span> <span m="1505700">other</span> <span m="1505860">five</span> <span m="1506200">cases.</span> <span m="1507810">The</span> <span m="1507890">probability</span> <span m="1508160">T</span> <span m="1508430">equals</span> <span m="1509200">0</span> <span m="1510910">given</span> <span m="1511290">any</span> <span m="1511590">value</span> <span m="1511950">of</span> <span m="1512040">D</span> <span m="1514200">is</span> <span m="1514410">5/6,</span> <span m="1516050">which</span> <span m="1516290">equals,</span> <span m="1516690">the</span> <span m="1516780">probability</span> <span m="1517280">T</span> <span m="1517500">equals</span> <span m="1517780">0.</span> <span m="1519910">So</span> <span m="1520080">we</span> <span m="1520220">check</span> <span m="1520690">all</span> <span m="1521060">possible</span> <span m="1521590">values</span> <span m="1521980">of</span> <span m="1522070">T1</span> <span m="1522860">and</span> <span m="1522980">all</span> <span m="1523160">possible</span> <span m="1523560">ways</span> <span m="1523730">of</span> <span m="1523830">D1,</span> <span m="1524213">and</span> <span m="1524980">lo</span> <span m="1525170">and</span> <span m="1525250">behold,</span> <span m="1525710">it</span> <span m="1525810">works</span> <span m="1525980">every</span> <span m="1526190">time.</span> </p>
<p><span m="1528050">So</span> <span m="1528210">knowing</span> <span m="1528550">the</span> <span m="1528640">first</span> <span m="1528970">die</span> <span m="1529210">does</span> <span m="1529410">not</span> <span m="1529840">change</span> <span m="1530210">the</span> <span m="1530280">probability</span> <span m="1530940">of</span> <span m="1531040">getting</span> <span m="1531490">a</span> <span m="1531590">7.</span> <span m="1533630">Would</span> <span m="1533770">it</span> <span m="1533850">change</span> <span m="1534140">the</span> <span m="1534230">probability</span> <span m="1534810">of</span> <span m="1534910">getting</span> <span m="1535220">a</span> <span m="1535280">6?</span> <span m="1538150">Yeah,</span> <span m="1539230">because</span> <span m="1539470">if</span> <span m="1539670">the</span> <span m="1539750">first</span> <span m="1540030">die</span> <span m="1540220">were</span> <span m="1540450">a</span> <span m="1540520">6,</span> <span m="1541040">you</span> <span m="1541150">know</span> <span m="1541280">you're</span> <span m="1541410">going</span> <span m="1541500">to</span> <span m="1541540">get</span> <span m="1541690">more</span> <span m="1541890">than a</span> <span m="1542100">6.</span> <span m="1542960">So</span> <span m="1543110">it</span> <span m="1543200">just</span> <span m="1543550">holds</span> <span m="1543860">true</span> <span m="1544090">because</span> <span m="1544400">I</span> <span m="1544500">picked</span> <span m="1546000">it</span> <span m="1546150">to</span> <span m="1546230">be</span> <span m="1546360">an</span> <span m="1546440">indicator</span> <span m="1547330">for</span> <span m="1547470">7.</span> <span m="1549220">Any</span> <span m="1549420">questions</span> <span m="1552210">about</span> <span m="1552450">that?</span> </p>
<p><span m="1552780">So</span> <span m="1553890">if</span> <span m="1554020">you're</span> <span m="1554180">ever</span> <span m="1554390">asked</span> <span m="1554650">to</span> <span m="1554700">show</span> <span m="1554970">things</span> <span m="1555210">are</span> <span m="1555340">independent,</span> <span m="1556900">you've</span> <span m="1557000">got</span> <span m="1557140">to</span> <span m="1557190">go</span> <span m="1557320">through</span> <span m="1557470">all</span> <span m="1557630">the</span> <span m="1557720">cases.</span> <span m="1559450">If</span> <span m="1559590">you're</span> <span m="1559760">asked</span> <span m="1559950">to</span> <span m="1560020">show</span> <span m="1560230">their</span> <span m="1560380">dependent,</span> <span m="1561090">just</span> <span m="1561340">find</span> <span m="1561580">one</span> <span m="1561790">case</span> <span m="1562130">that</span> <span m="1562210">blows it</span> <span m="1562560">up,</span> <span m="1564120">and</span> <span m="1564230">you're</span> <span m="1564330">done.</span> <span m="1566830">OK,</span> <span m="1567980">you</span> <span m="1568410">can</span> <span m="1568560">also</span> <span m="1568810">talk</span> <span m="1569060">about</span> <span m="1569310">independence</span> <span m="1570090">for</span> <span m="1570870">many</span> <span m="1571640">random</span> <span m="1571990">variables.</span> <span m="1572480">And</span> <span m="1572540">you</span> <span m="1572620">have</span> <span m="1572780">a</span> <span m="1572840">notion</span> <span m="1573190">of</span> <span m="1573300">mutual</span> <span m="1573980">independence.</span> <span m="1586900">And</span> <span m="1587040">this</span> <span m="1587190">is</span> <span m="1587530">a</span> <span m="1587610">little</span> <span m="1587890">hairy.</span> <span m="1590080">It's a</span> <span m="1590170">natural</span> <span m="1590710">thing.</span> <span m="1592090">R1,</span> <span m="1593560">R2,</span> <span m="1595360">up</span> <span m="1595530">to</span> <span m="1595640">Rn</span> <span m="1597632">are</span> <span m="1598130">mutually</span> <span m="1598750">independent</span> <span m="1611050">if</span> <span m="1613740">for</span> <span m="1613990">all</span> <span m="1614400">the</span> <span m="1614490">values</span> <span m="1615110">of</span> <span m="1615210">all</span> <span m="1615620">the</span> <span m="1615740">random</span> <span m="1616110">variables,</span> <span m="1616940">so</span> <span m="1617350">x1,</span> <span m="1618690">x2,</span> <span m="1620150">xn.</span> <span m="1624150">And</span> <span m="1624320">we're</span> <span m="1624430">going</span> <span m="1624530">to</span> <span m="1624570">give</span> <span m="1624710">the</span> <span m="1624810">product</span> <span m="1625320">form,</span> <span m="1625840">because</span> <span m="1626010">it's</span> <span m="1626190">the</span> <span m="1626250">simpler</span> <span m="1626640">way</span> <span m="1626770">to</span> <span m="1626880">do</span> <span m="1627110">it</span> <span m="1627170">here.</span> </p>
<p><span m="1628650">The</span> <span m="1628740">probability</span> <span m="1629420">that</span> <span m="1629580">R1</span> <span m="1630520">is</span> <span m="1630740">x1,</span> <span m="1631660">and</span> <span m="1632660">R2</span> <span m="1633290">is</span> <span m="1633430">x2,</span> <span m="1634510">and</span> <span m="1636610">Rn</span> <span m="1637120">is</span> <span m="1637250">xn</span> <span m="1639170">equals</span> <span m="1639560">the</span> <span m="1639660">product</span> <span m="1640330">of</span> <span m="1640390">the</span> <span m="1640510">individual</span> <span m="1640970">probabilities.</span> <span m="1650880">It's</span> <span m="1651250">the</span> <span m="1651330">natural</span> <span m="1652200">generalization</span> <span m="1654380">for</span> <span m="1654520">two</span> <span m="1654710">variables.</span> <span m="1655420">And</span> <span m="1655590">there's</span> <span m="1655770">also</span> <span m="1656200">the</span> <span m="1656350">equivalent</span> <span m="1657000">form</span> <span m="1657950">in</span> <span m="1658100">conditional</span> <span m="1658640">probabilities,</span> <span m="1659700">but</span> <span m="1659830">that's</span> <span m="1660030">a</span> <span m="1660070">little</span> <span m="1660430">harrier to</span> <span m="1660900">write</span> <span m="1661130">down,</span> <span m="1661400">and</span> <span m="1661460">we</span> <span m="1661560">generally</span> <span m="1662680">don't</span> <span m="1663260">work</span> <span m="1663520">with</span> <span m="1663610">that.</span> <span m="1669880">Any</span> <span m="1670050">questions</span> <span m="1670560">about</span> <span m="1671900">mutual</span> <span m="1672470">independence?</span> <span m="1674340">Yeah.</span> </p>
<p><span m="1675227">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1684173">PROFESSOR: So any</span> <span m="1684670">subset</span> <span m="1685167">what?</span> </p>
<p><span m="1685664">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1689860">PROFESSOR: Yes,</span> <span m="1690620">in</span> <span m="1690790">the</span> <span m="1691030">conditional</span> <span m="1691510">form</span> <span m="1691720">you</span> <span m="1691800">check</span> <span m="1692050">any</span> <span m="1692220">subset,</span> <span m="1693730">here</span> <span m="1695450">you don't</span> <span m="1695630">have</span> <span m="1695770">to</span> <span m="1695850">worry</span> <span m="1696030">about</span> <span m="1696190">the</span> <span m="1696290">subsets.</span> <span m="1696940">You</span> <span m="1697040">take</span> <span m="1697470">any</span> <span m="1697840">possible</span> <span m="1698400">value</span> <span m="1698950">for</span> <span m="1699900">all</span> <span m="1700280">of</span> <span m="1700360">them,</span> <span m="1700550">so</span> <span m="1700840">there's</span> <span m="1700930">no</span> <span m="1701100">subset</span> <span m="1701630">notation</span> <span m="1702130">in</span> <span m="1702360">this</span> <span m="1702630">version,</span> <span m="1704290">so</span> <span m="1704430">you</span> <span m="1704940">can</span> <span m="1705240">do</span> <span m="1705370">without</span> <span m="1705490">it.</span> <span m="1706010">That</span> <span m="1706140">would</span> <span m="1706250">be</span> <span m="1706390">equivalent,</span> <span m="1706950">there's</span> <span m="1707080">another</span> <span m="1707310">form</span> <span m="1707660">where you</span> <span m="1707800">look</span> <span m="1707940">at</span> <span m="1707990">all the</span> <span m="1708260">possible</span> <span m="1708620">subsets</span> <span m="1709130">as</span> <span m="1709240">well, and</span> <span m="1709600">conditioning</span> <span m="1710090">on</span> <span m="1710200">that.</span> <span m="1711030">But</span> <span m="1711210">this</span> <span m="1711420">is</span> <span m="1711540">good</span> <span m="1711690">enough,</span> <span m="1712330">and</span> <span m="1712750">this</span> <span m="1712910">is</span> <span m="1713000">probably</span> <span m="1713210">the</span> <span m="1713300">simplest</span> <span m="1713820">way</span> <span m="1713920">to</span> <span m="1714000">do the</span> <span m="1714190">mutual</span> <span m="1714580">independence,</span> <span m="1714890">is</span> <span m="1715200">like</span> <span m="1715400">this.</span> <span m="1716150">So</span> <span m="1716390">every</span> <span m="1716700">variable</span> <span m="1716970">is</span> <span m="1717240">included</span> <span m="1717650">here.</span> </p>
<p><span m="1718850">And</span> <span m="1718970">that</span> <span m="1719140">will</span> <span m="1719280">imply</span> <span m="1720320">the</span> <span m="1720410">same</span> <span m="1720740">thing</span> <span m="1720950">for</span> <span m="1721080">any</span> <span m="1721230">subset.</span> <span m="1721950">You</span> <span m="1722050">don't</span> <span m="1722170">have</span> <span m="1722250">to</span> <span m="1722340">check</span> <span m="1722660">every</span> <span m="1722810">subset</span> <span m="1723240">here.</span> <span m="1725380">And</span> <span m="1725490">then</span> <span m="1725660">you</span> <span m="1725860">could</span> <span m="1726010">do</span> <span m="1726130">that</span> <span m="1726330">by</span> <span m="1726420">just</span> <span m="1727150">summing</span> <span m="1727620">over</span> <span m="1727960">Rn</span> <span m="1728390">being</span> <span m="1728640">everything,</span> <span m="1729395">and</span> <span m="1729660">that</span> <span m="1730290">cancels</span> <span m="1730700">away</span> <span m="1731560">to</span> <span m="1732060">get</span> <span m="1732220">rid</span> <span m="1732360">of</span> <span m="1732470">items.</span> <span m="1733530">But</span> <span m="1733650">you</span> <span m="1733700">don't</span> <span m="1733830">have</span> <span m="1733920">to</span> <span m="1734010">worry</span> <span m="1734180">about</span> <span m="1734410">that.</span> </p>
<p><span m="1738020">OK,</span> <span m="1738840">so</span> <span m="1739000">we're going to</span> <span m="1739430">change</span> <span m="1739780">gears</span> <span m="1740100">a</span> <span m="1740160">little</span> <span m="1740360">bit</span> <span m="1740540">now,</span> <span m="1741720">and</span> <span m="1742830">talk</span> <span m="1743140">about</span> <span m="1744050">the</span> <span m="1744900">probability</span> <span m="1746220">distribution</span> <span m="1746990">function,</span> <span m="1748230">which</span> <span m="1748580">is</span> <span m="1748730">just</span> <span m="1749010">a</span> <span m="1749120">way</span> <span m="1749390">a</span> <span m="1749460">sort</span> <span m="1749690">of</span> <span m="1750220">writing</span> <span m="1750600">down</span> <span m="1750900">or</span> <span m="1750970">characterizing</span> <span m="1752410">the</span> <span m="1752520">probabilities</span> <span m="1753660">associated</span> <span m="1754860">with</span> <span m="1755150">each</span> <span m="1755670">value</span> <span m="1756090">being</span> <span m="1756350">attained.</span> <span m="1761920">So</span> <span m="1762130">given</span> <span m="1763820">a</span> <span m="1763910">random</span> <span m="1764340">variable,</span> <span m="1765330">R,</span> <span m="1769380">the</span> <span m="1769610">probability</span> <span m="1773690">also</span> <span m="1774150">known as</span> <span m="1774550">the</span> <span m="1774800">point,</span> <span m="1780140">distribution</span> <span m="1780970">function,</span> <span m="1793360">also</span> <span m="1793920">denoted</span> <span m="1794480">pdf,</span> <span m="1795450">probability</span> <span m="1795920">distribution</span> <span m="1796510">function,</span> <span m="1799340">for</span> <span m="1799780">R</span> <span m="1801580">is,</span> <span m="1802560">well, it's</span> <span m="1803020">very</span> <span m="1803220">simple.</span> <span m="1803700">It's just</span> <span m="1803940">the</span> <span m="1804020">function</span> <span m="1805220">f</span> <span m="1805450">of</span> <span m="1805580">x,</span> <span m="1806630">which</span> <span m="1806850">equals</span> <span m="1807940">the</span> <span m="1808010">probability</span> <span m="1808620">that</span> <span m="1809280">R</span> <span m="1809600">is</span> <span m="1809910">x.</span> <span m="1811320">Very</span> <span m="1811650">simple.</span> <span m="1812940">But</span> <span m="1813070">now</span> <span m="1813310">we're</span> <span m="1813730">characterizing</span> <span m="1814560">it</span> <span m="1814650">as</span> <span m="1815360">a</span> <span m="1815440">function.</span> </p>
<p><span m="1817170">And</span> <span m="1817290">there's</span> <span m="1817440">also</span> <span m="1817700">a</span> <span m="1817740">notion</span> <span m="1818060">of</span> <span m="1818150">the</span> <span m="1818260">cumulative</span> <span m="1819470">distribution</span> <span m="1820320">function,</span> <span m="1821590">which</span> <span m="1821970">is</span> <span m="1823270">going</span> <span m="1823400">to</span> <span m="1823440">be</span> <span m="1823550">the</span> <span m="1823650">probability</span> <span m="1824320">that</span> <span m="1824500">R</span> <span m="1824830">is</span> <span m="1824980">less</span> <span m="1825280">than</span> <span m="1825460">or</span> <span m="1825540">equal</span> <span m="1825770">to</span> <span m="1825840">x.</span> <span m="1827130">Let's</span> <span m="1827330">write</span> <span m="1827530">that</span> <span m="1827740">down.</span> <span m="1852800">The</span> <span m="1852980">cumulative</span> <span m="1864190">distribution</span> <span m="1864970">function</span> <span m="1869524">F</span> <span m="1872500">for</span> <span m="1872840">random</span> <span m="1873160">variable</span> <span m="1873690">R</span> <span m="1875780">is</span> <span m="1876170">simply</span> <span m="1879550">f</span> <span m="1879820">of</span> <span m="1879960">x</span> <span m="1881030">is</span> <span m="1881400">the</span> <span m="1881500">probability</span> <span m="1882230">that</span> <span m="1882410">R</span> <span m="1883380">is</span> <span m="1883870">at</span> <span m="1883940">most</span> <span m="1884280">x,</span> <span m="1885250">which</span> <span m="1885460">is</span> <span m="1885570">just</span> <span m="1885840">the</span> <span m="1885930">sum</span> <span m="1887650">over</span> <span m="1887950">all</span> <span m="1888280">y</span> <span m="1888750">less</span> <span m="1889040">than</span> <span m="1889160">or</span> <span m="1889180">equal</span> <span m="1889500">to</span> <span m="1889580">x</span> <span m="1890730">the</span> <span m="1890820">probability</span> <span m="1891510">R</span> <span m="1891740">equals</span> <span m="1892010">y.</span> </p>
<p><span m="1895480">So</span> <span m="1896990">the</span> <span m="1897420">distribution</span> <span m="1898090">functions</span> <span m="1898590">characterize</span> <span m="1900230">the</span> <span m="1900340">probability</span> <span m="1900930">a</span> <span m="1901000">random</span> <span m="1901320">variable</span> <span m="1901710">takes</span> <span m="1902020">on</span> <span m="1902160">any</span> <span m="1902370">value,</span> <span m="1903560">and</span> <span m="1903830">there's</span> <span m="1904010">certain</span> <span m="1904530">common</span> <span m="1905110">ones that</span> <span m="1905530">just</span> <span m="1905740">come</span> <span m="1905900">up</span> <span m="1906040">all</span> <span m="1906230">the</span> <span m="1906330">time.</span> <span m="1907400">And</span> <span m="1907470">so</span> <span m="1907600">people</span> <span m="1907950">have</span> <span m="1908050">done</span> <span m="1908140">a</span> <span m="1908210">lot</span> <span m="1908560">of</span> <span m="1908640">analysis</span> <span m="1909190">about</span> <span m="1909580">them</span> <span m="1910940">because</span> <span m="1911510">they</span> <span m="1911650">occur</span> <span m="1911960">so</span> <span m="1912140">frequently.</span> <span m="1912990">And</span> <span m="1913100">we're</span> <span m="1913180">going</span> <span m="1913260">to</span> <span m="1913330">talk</span> <span m="1913610">about</span> <span m="1914210">three</span> <span m="1914660">of</span> <span m="1914770">them</span> <span m="1915710">today.</span> <span m="1922830">The</span> <span m="1923320">first</span> <span m="1923760">is</span> <span m="1923870">really</span> <span m="1924180">simple,</span> <span m="1925660">and</span> <span m="1926580">that</span> <span m="1926860">is</span> <span m="1928190">the</span> <span m="1928420">Bernoulli</span> <span m="1929060">random</span> <span m="1929420">variable,</span> <span m="1930000">or</span> <span m="1930240">indicator</span> <span m="1931100">random</span> <span m="1931430">variable.</span> </p>
<p><span m="1934430">All</span> <span m="1934510">right,</span> <span m="1934750">so</span> <span m="1934970">for</span> <span m="1937430">an</span> <span m="1937640">indicator</span> <span m="1942200">random</span> <span m="1942540">variable,</span> <span m="1945760">f</span> <span m="1946450">0</span> <span m="1947880">is</span> <span m="1948150">p,</span> <span m="1949520">and</span> <span m="1949940">f</span> <span m="1950900">1</span> <span m="1951860">is</span> <span m="1952030">1</span> <span m="1952230">minus</span> <span m="1952640">p,</span> <span m="1953110">for</span> <span m="1953270">sum</span> <span m="1953570">p.</span> <span m="1956290">The</span> <span m="1956400">probability</span> <span m="1956900">could</span> <span m="1957050">be</span> <span m="1957375">half,</span> <span m="1957700">if</span> <span m="1957840">you're</span> <span m="1958020">flipping</span> <span m="1958330">a</span> <span m="1958400">coin.</span> <span m="1958860">The</span> <span m="1959250">probability</span> <span m="1959680">of</span> <span m="1959740">heads</span> <span m="1960020">could</span> <span m="1960110">be a</span> <span m="1960240">half,</span> <span m="1960720">probability of</span> <span m="1961060">tails</span> <span m="1961400">would</span> <span m="1961490">be</span> <span m="1961620">a half.</span> <span m="1962030">It's</span> <span m="1962120">just</span> <span m="1962800">two</span> <span m="1962990">values,</span> <span m="1963500">0</span> <span m="1963810">and</span> <span m="1963970">1.</span> <span m="1965360">And</span> <span m="1965510">then</span> <span m="1966200">the</span> <span m="1966500">cumulative</span> <span m="1966630">function</span> <span m="1967090">is</span> <span m="1967260">very</span> <span m="1967510">simple.</span> <span m="1968770">F</span> <span m="1968950">0</span> <span m="1969280">is</span> <span m="1969610">p,</span> <span m="1970870">big</span> <span m="1971090">F</span> <span m="1971300">of</span> <span m="1971490">1</span> <span m="1972520">is</span> <span m="1972690">1.</span> <span m="1975710">So</span> <span m="1975810">that</span> <span m="1975980">is</span> <span m="1976300">about</span> <span m="1976480">as</span> <span m="1976530">simple</span> <span m="1976820">of</span> <span m="1977110">functions</span> <span m="1977530">as you</span> <span m="1977800">can</span> <span m="1977950">get.</span> </p>
<p><span m="1983670">The</span> <span m="1983790">next</span> <span m="1984450">simplest,</span> <span m="1986210">and</span> <span m="1986360">also</span> <span m="1986610">very</span> <span m="1986900">common,</span> <span m="1987390">is</span> <span m="1987520">called</span> <span m="1987840">a</span> <span m="1987900">uniform</span> <span m="1989540">random</span> <span m="1989890">variable.</span> <span m="1994970">Let's</span> <span m="1995430">define</span> <span m="1996120">that.</span> <span m="2006300">For</span> <span m="2006490">a</span> <span m="2006540">uniform</span> <span m="2011070">random</span> <span m="2011450">variable--</span> <span m="2012410">and</span> <span m="2012510">we</span> <span m="2012610">have</span> <span m="2012710">to</span> <span m="2012810">define</span> <span m="2013350">what</span> <span m="2013530">it's</span> <span m="2013610">defined</span> <span m="2013945">on--</span> <span m="2015000">on</span> <span m="2015310">say</span> <span m="2015620">the</span> <span m="2015980">integers</span> <span m="2016500">from</span> <span m="2016690">1</span> <span m="2016960">to</span> <span m="2017110">n,</span> <span m="2019440">every</span> <span m="2019830">value</span> <span m="2021020">is</span> <span m="2021200">equally</span> <span m="2021540">likely.</span> <span m="2022900">All</span> <span m="2023170">the</span> <span m="2023280">integers</span> <span m="2023570">from</span> <span m="2023690">1</span> <span m="2023920">to</span> <span m="2023970">n</span> <span m="2024180">are</span> <span m="2024280">equally</span> <span m="2024630">likely</span> <span m="2024950">to</span> <span m="2025040">occur.</span> <span m="2026600">And</span> <span m="2026820">so</span> <span m="2027110">in</span> <span m="2027180">this</span> <span m="2027370">case,</span> <span m="2028260">n</span> <span m="2028470">is</span> <span m="2028580">a</span> <span m="2028670">parameter</span> <span m="2029330">of</span> <span m="2029400">the</span> <span m="2029490">function.</span> <span m="2031470">fn</span> <span m="2031910">of</span> <span m="2032030">K</span> <span m="2032540">is</span> <span m="2032680">simply</span> <span m="2033050">1/n.</span> <span m="2034750">Each</span> <span m="2035120">integer</span> <span m="2035530">from</span> <span m="2035690">1</span> <span m="2035900">to</span> <span m="2035960">n</span> <span m="2036210">is</span> <span m="2036330">equally</span> <span m="2036650">likely.</span> <span m="2037860">So</span> <span m="2037940">it's</span> <span m="2038050">1</span> <span m="2038860">and</span> <span m="2038920">n.</span> </p>
<p><span m="2043990">All</span> <span m="2044070">right,</span> <span m="2044220">what</span> <span m="2044490">is</span> <span m="2044880">the</span> <span m="2045020">cumulative</span> <span m="2045700">distribution</span> <span m="2046360">function</span> <span m="2046800">for</span> <span m="2046930">this?</span> <span m="2048920">What</span> <span m="2049130">is</span> <span m="2049300">big</span> <span m="2049639">F</span> <span m="2049830">of</span> <span m="2049960">n</span> <span m="2050790">of</span> <span m="2050940">K</span> <span m="2053320">for</span> <span m="2053400">the</span> <span m="2053530">uniform?</span> <span m="2053989">The</span> <span m="2054080">probability</span> <span m="2054969">that</span> <span m="2055179">the</span> <span m="2056340">random</span> <span m="2056800">variable</span> <span m="2058350">takes</span> <span m="2058650">a</span> <span m="2058690">value</span> <span m="2059179">that's</span> <span m="2059440">at</span> <span m="2059610">most</span> <span m="2060150">K?</span> </p>
<p><span m="2062518">AUDIENCE: [INAUDIBLE]?</span> </p>
<p><span m="2064449">PROFESSOR: Close,</span> <span m="2065159">K/n.</span> <span m="2067199">And</span> <span m="2067370">you</span> <span m="2067460">could</span> <span m="2067590">think</span> <span m="2067790">of</span> <span m="2067900">K</span> <span m="2068199">being</span> <span m="2068489">1.</span> <span m="2070139">You</span> <span m="2070270">have</span> <span m="2070480">a</span> <span m="2070540">least</span> <span m="2070989">the</span> <span m="2071100">1</span> <span m="2071320">over n</span> <span m="2071639">probability.</span> <span m="2073469">So</span> <span m="2073560">you</span> <span m="2073639">have</span> <span m="2073800">K</span> <span m="2074100">chances.</span> <span m="2075290">It</span> <span m="2075370">could</span> <span m="2075480">be</span> <span m="2075570">1,</span> <span m="2076080">2,</span> <span m="2076409">3,</span> <span m="2076679">4</span> <span m="2077000">up</span> <span m="2077159">to</span> <span m="2077260">K,</span> <span m="2077659">each</span> <span m="2077880">has</span> <span m="2078010">a</span> <span m="2078090">1/n</span> <span m="2078630">chance.</span> <span m="2080159">Because</span> <span m="2080620">we've</span> <span m="2080800">got,</span> <span m="2081639">the</span> <span m="2081730">definition</span> <span m="2082380">is</span> <span m="2084850">less</span> <span m="2085130">than</span> <span m="2085310">or</span> <span m="2085429">equal,</span> <span m="2086790">not</span> <span m="2087010">less</span> <span m="2087210">than.</span> <span m="2091020">Uniform</span> <span m="2091500">distributions</span> <span m="2092090">come</span> <span m="2092230">up</span> <span m="2092360">all</span> <span m="2092510">the</span> <span m="2092610">time,</span> <span m="2093250">rolling</span> <span m="2093830">a</span> <span m="2093929">fair</span> <span m="2094250">die</span> <span m="2095530">is</span> <span m="2095679">uniform</span> <span m="2096489">on</span> <span m="2096719">1</span> <span m="2096969">through</span> <span m="2097130">6</span> <span m="2098040">for</span> <span m="2098230">values.</span> <span m="2099460">If I</span> <span m="2099570">pick</span> <span m="2099770">a</span> <span m="2099890">random</span> <span m="2100250">student</span> <span m="2100580">in</span> <span m="2100650">the</span> <span m="2100730">class,</span> <span m="2102120">the</span> <span m="2102210">chance</span> <span m="2102560">I</span> <span m="2102640">pick</span> <span m="2102900">you</span> <span m="2103200">is</span> <span m="2103500">1</span> <span m="2103910">in</span> <span m="2104420">the</span> <span m="2104510">size</span> <span m="2104830">of</span> <span m="2104900">the</span> <span m="2104990">class.</span> <span m="2105940">We</span> <span m="2106040">have</span> <span m="2106140">uniform</span> <span m="2106590">sample</span> <span m="2107010">spaces.</span> <span m="2107520">Each</span> <span m="2107780">outcome</span> <span m="2108210">occurs</span> <span m="2109170">equally</span> <span m="2109560">likely.</span> <span m="2112300">All</span> <span m="2112440">right,</span> <span m="2112920">any</span> <span m="2113150">questions</span> <span m="2113670">about</span> <span m="2114020">uniform?</span> <span m="2116340">[INAUDIBLE].</span> </p>
<p><span m="2118600">OK,</span> <span m="2118940">so</span> <span m="2119120">while</span> <span m="2119310">we're</span> <span m="2119410">talking</span> <span m="2119640">about</span> <span m="2119840">uniform,</span> <span m="2120260">I</span> <span m="2120330">want</span> <span m="2120540">to</span> <span m="2120610">play</span> <span m="2121340">a</span> <span m="2121450">game,</span> <span m="2123400">whose</span> <span m="2124500">optimal</span> <span m="2124970">strategy</span> <span m="2125530">is</span> <span m="2125640">going</span> <span m="2125740">to</span> <span m="2125800">turn</span> <span m="2126030">out</span> <span m="2126180">to</span> <span m="2126250">be</span> <span m="2126370">related</span> <span m="2126980">to</span> <span m="2127070">uniform</span> <span m="2127560">distributions.</span> <span m="2130120">Now</span> <span m="2130500">the</span> <span m="2130630">game</span> <span m="2130950">works</span> <span m="2131210">as</span> <span m="2131340">follows.</span> <span m="2133750">I'm</span> <span m="2133810">going</span> <span m="2133910">to</span> <span m="2133950">have</span> <span m="2134800">two</span> <span m="2135020">envelopes</span> <span m="2135560">here.</span> <span m="2137890">And</span> <span m="2138150">inside</span> <span m="2139460">each</span> <span m="2139710">envelope,</span> <span m="2140380">I've</span> <span m="2140570">written</span> <span m="2140810">a</span> <span m="2140870">number.</span> <span m="2142400">And</span> <span m="2142700">the</span> <span m="2142770">number</span> <span m="2143150">in</span> <span m="2143220">this</span> <span m="2143430">case</span> <span m="2144120">is</span> <span m="2144690">a</span> <span m="2144770">number</span> <span m="2145060">between</span> <span m="2145490">0</span> <span m="2145775">and</span> <span m="2146060">100</span> <span m="2146860">inclusive.</span> <span m="2150710">It's</span> <span m="2150820">not</span> <span m="2151010">a</span> <span m="2151070">random</span> <span m="2151430">number,</span> <span m="2152260">because</span> <span m="2152450">I</span> <span m="2152590">wrote</span> <span m="2152890">it,</span> <span m="2154090">and</span> <span m="2155210">maybe</span> <span m="2155310">I'm not</span> <span m="2155600">such</span> <span m="2155770">a</span> <span m="2155820">nice</span> <span m="2156060">guy.</span> <span m="2156600">I</span> <span m="2156660">might</span> <span m="2156860">have</span> <span m="2156940">picked</span> <span m="2157210">nasty</span> <span m="2157580">numbers</span> <span m="2157950">to</span> <span m="2158060">write</span> <span m="2158260">here.</span> <span m="2159230">They</span> <span m="2159410">are</span> <span m="2159590">different,</span> <span m="2160020">though.</span> <span m="2160910">The</span> <span m="2161010">numbers</span> <span m="2161430">are</span> <span m="2161560">different</span> <span m="2161980">in</span> <span m="2162100">the</span> <span m="2162190">envelopes.</span> </p>
<p><span m="2164060">Now</span> <span m="2165460">I'm</span> <span m="2165590">going</span> <span m="2165690">to</span> <span m="2165780">pick</span> <span m="2165970">a</span> <span m="2166030">volunteer</span> <span m="2166510">from</span> <span m="2166620">the</span> <span m="2166720">class,</span> <span m="2167260">and</span> <span m="2167490">their</span> <span m="2167690">job</span> <span m="2168920">is</span> <span m="2169090">to</span> <span m="2169220">pick</span> <span m="2169580">the</span> <span m="2169740">envelope</span> <span m="2170180">with</span> <span m="2170280">the</span> <span m="2170350">bigger</span> <span m="2170630">number.</span> <span m="2172200">And</span> <span m="2172440">if</span> <span m="2172520">they</span> <span m="2172650">get</span> <span m="2172930">the one</span> <span m="2173070">with</span> <span m="2173250">the</span> <span m="2173320">bigger</span> <span m="2173570">number,</span> <span m="2174910">then</span> <span m="2175110">they</span> <span m="2175230">get</span> <span m="2176670">one</span> <span m="2176860">of</span> <span m="2176910">these</span> <span m="2177130">guys</span> <span m="2177975">for</span> <span m="2178350">ice</span> <span m="2178580">cream.</span> <span m="2179520">And</span> <span m="2179800">if</span> <span m="2179910">they</span> <span m="2180080">don't</span> <span m="2180400">get</span> <span m="2180550">the</span> <span m="2180660">one</span> <span m="2180820">with</span> <span m="2180860">the</span> <span m="2180940">bigger</span> <span m="2181190">number,</span> <span m="2182910">they</span> <span m="2183070">get</span> <span m="2183650">the</span> <span m="2183770">nerd</span> <span m="2184060">pride</span> <span m="2184400">pocket</span> <span m="2184720">protector.</span> </p>
<p><span m="2187340">Now,</span> <span m="2188420">since</span> <span m="2188620">you</span> <span m="2188690">don't</span> <span m="2188960">know</span> <span m="2189210">which</span> <span m="2189840">number</span> <span m="2190080">I</span> <span m="2190110">put</span> <span m="2190310">in</span> <span m="2190430">which,</span> <span m="2191030">50-50</span> <span m="2191370">if</span> <span m="2191810">you</span> <span m="2191890">get</span> <span m="2192070">the</span> <span m="2192150">prize,</span> <span m="2192650">the</span> <span m="2193230">$10</span> <span m="2194180">gift</span> <span m="2194360">certificate.</span> <span m="2195520">So</span> <span m="2195670">to</span> <span m="2195740">give</span> <span m="2195930">you</span> <span m="2196040">a</span> <span m="2196100">little</span> <span m="2196390">bit</span> <span m="2196530">of</span> <span m="2196610">an</span> <span m="2196710">edge,</span> <span m="2197980">I'm</span> <span m="2198070">going</span> <span m="2198200">to</span> <span m="2198270">let</span> <span m="2198660">you</span> <span m="2198980">open</span> <span m="2199420">one</span> <span m="2199660">of</span> <span m="2199720">the</span> <span m="2199840">envelopes</span> <span m="2201060">and</span> <span m="2201240">see</span> <span m="2201480">what</span> <span m="2201660">number's</span> <span m="2202010">there.</span> <span m="2203320">Then</span> <span m="2203960">you</span> <span m="2204120">have</span> <span m="2204260">to</span> <span m="2204360">decide,</span> <span m="2205560">do</span> <span m="2205660">you</span> <span m="2205750">want</span> <span m="2205980">the</span> <span m="2206060">number</span> <span m="2206310">you</span> <span m="2206440">got,</span> <span m="2206780">or</span> <span m="2206930">do</span> <span m="2206990">you</span> <span m="2207090">want</span> <span m="2207320">the</span> <span m="2207430">other</span> <span m="2207630">envelope?</span> <span m="2210210">OK?</span> <span m="2212330">Is</span> <span m="2212450">the</span> <span m="2212530">game</span> <span m="2212950">clear,</span> <span m="2213895">what</span> <span m="2214360">we're</span> <span m="2214520">going</span> <span m="2214600">to</span> <span m="2214650">do?</span> </p>
<p><span m="2214850">OK,</span> <span m="2215220">so</span> <span m="2215670">need</span> <span m="2216070">a</span> <span m="2216100">couple</span> <span m="2216360">of</span> <span m="2216440">volunteers.</span> <span m="2216740">We're</span> <span m="2217040">going to</span> <span m="2217140">do</span> <span m="2217280">it</span> <span m="2217380">twice.</span> <span m="2218950">All right,</span> <span m="2219090">somebody</span> <span m="2219380">way</span> <span m="2219600">in</span> <span m="2219680">the</span> <span m="2219760">back up</span> <span m="2220170">there.</span> <span m="2221590">Any</span> <span m="2221800">other</span> <span m="2221950">volunteers?</span> <span m="2222520">All right,</span> <span m="2222750">come</span> <span m="2222970">on</span> <span m="2223090">down.</span> <span m="2223390">Oh,</span> <span m="2223500">you've</span> <span m="2223610">already</span> <span m="2223910">played</span> <span m="2224140">before.</span> <span m="2225017">(LAUGHTER)</span> <span m="2226730">You already</span> <span m="2227060">got.</span> <span m="2227190">Who</span> <span m="2227330">hasn't</span> <span m="2227600">played</span> <span m="2227870">before,</span> <span m="2228840">wants</span> <span m="2229050">to--</span> <span m="2229280">all right,</span> <span m="2229550">come</span> <span m="2229770">on</span> <span m="2229930">up.</span> <span m="2232510">Now</span> <span m="2232810">you</span> <span m="2233000">want</span> <span m="2233140">to</span> <span m="2233180">be</span> <span m="2233320">thinking</span> <span m="2233700">about,</span> <span m="2234730">boy,</span> <span m="2234900">is</span> <span m="2235030">there</span> <span m="2235200">a</span> <span m="2235260">strategy</span> <span m="2235870">here?</span> <span m="2236140">Or</span> <span m="2236300">is</span> <span m="2236390">this</span> <span m="2236570">just</span> <span m="2236840">dumb</span> <span m="2237100">luck,</span> <span m="2237360">50-50?</span> </p>
<p><span m="2240960">OK,</span> <span m="2241290">what's</span> <span m="2241460">your</span> <span m="2241550">name?</span> </p>
<p><span m="2242020">AUDIENCE: Sean.</span> </p>
<p><span m="2242590">PROFESSOR: Sean.</span> <span m="2243410">All right,</span> <span m="2243670">I</span> <span m="2243900">got</span> <span m="2244100">two</span> <span m="2244240">envelopes,</span> <span m="2245200">Sean.</span> <span m="2245700">They are</span> <span m="2245930">numbers</span> <span m="2246280">between</span> <span m="2246580">0</span> <span m="2246910">and</span> <span m="2246970">100.</span> <span m="2248170">And</span> <span m="2250000">you</span> <span m="2250170">can</span> <span m="2250860">pick</span> <span m="2251090">one,</span> <span m="2251560">and</span> <span m="2251930">we'll</span> <span m="2252060">reveal</span> <span m="2252540">the</span> <span m="2252640">number,</span> <span m="2253910">and</span> <span m="2254030">then</span> <span m="2254160">you</span> <span m="2254250">decide.</span> <span m="2254540">Do</span> <span m="2254830">you</span> <span m="2254970">want</span> <span m="2255270">the</span> <span m="2255360">number</span> <span m="2255540">you</span> <span m="2255630">got, or</span> <span m="2255990">do</span> <span m="2256070">you</span> <span m="2256140">want</span> <span m="2256350">the</span> <span m="2256460">other</span> <span m="2256620">one?</span> <span m="2256890">The goal</span> <span m="2257210">is</span> <span m="2257290">to</span> <span m="2257390">get</span> <span m="2257610">the</span> <span m="2257670">bigger</span> <span m="2258070">number.</span> <span m="2259510">So</span> <span m="2259580">take</span> <span m="2259840">one and</span> <span m="2260090">open</span> <span m="2260330">it.</span> <span m="2268105">What did</span> <span m="2268602">you get?</span> </p>
<p><span m="2269099">AUDIENCE: 6.</span> <span m="2270590">He got</span> </p>
<p><span m="2271087">PROFESSOR: A 6.</span> <span m="2272590">The</span> <span m="2272700">numbers</span> <span m="2273340">go</span> <span m="2273780">from</span> <span m="2273990">0</span> <span m="2274700">to</span> <span m="2274870">100.</span> <span m="2276070">What</span> <span m="2276410">do</span> <span m="2276450">you</span> <span m="2276610">think</span> <span m="2276870">Sean</span> <span m="2277210">should</span> <span m="2277400">do?</span> </p>
<p><span m="2277880">AUDIENCE: Switch.</span> <span m="2279254">[INAUDIBLE].</span> </p>
<p><span m="2282250">PROFESSOR: What</span> <span m="2282630">do</span> <span m="2282680">you</span> <span m="2282810">think</span> <span m="2282990">you</span> <span m="2283070">should</span> <span m="2283250">do?</span> </p>
<p><span m="2284710">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="2288670">AUDIENCE: They've got</span> <span m="2289165">to check</span> <span m="2289660">the other</span> <span m="2290155">one.</span> </p>
<p><span m="2290650">PROFESSOR: No, no,</span> <span m="2290800">you</span> <span m="2291140">can't</span> <span m="2291390">look</span> <span m="2291550">at</span> <span m="2291620">the other</span> <span m="2292085">one.</span> <span m="2293480">And</span> <span m="2293700">unfortunately,</span> <span m="2294320">you're</span> <span m="2294400">going</span> <span m="2294500">to</span> <span m="2294540">play</span> <span m="2294710">with</span> <span m="2294820">different</span> <span m="2295110">envelopes.</span> <span m="2296850">What</span> <span m="2297240">should</span> <span m="2297380">you</span> <span m="2297655">do?</span> <span m="2298240">6,</span> <span m="2298650">there's</span> <span m="2298780">a</span> <span m="2298830">lot</span> <span m="2299130">of</span> <span m="2299180">numbers</span> <span m="2299490">bigger</span> <span m="2299740">than</span> <span m="2299890">6,</span> <span m="2300210">Sean.</span> <span m="2302410">But</span> <span m="2302570">they</span> <span m="2302660">might</span> <span m="2302880">not</span> <span m="2303040">be</span> <span m="2303130">in</span> <span m="2303210">that</span> <span m="2303380">envelope.</span> <span m="2305920">Might</span> <span m="2306130">be</span> <span m="2306210">a</span> <span m="2306350">0</span> <span m="2306680">in</span> <span m="2307010">that envelope.</span> </p>
<p><span m="2307730">AUDIENCE: That would suck.</span> </p>
<p><span m="2308900">PROFESSOR: Yeah.</span> </p>
<p><span m="2309180">AUDIENCE: I'll</span> <span m="2309645">stay.</span> </p>
<p><span m="2310110">PROFESSOR: You're going to</span> <span m="2310320">stay</span> <span m="2310520">with</span> <span m="2310660">6?</span> </p>
<p><span m="2311220">AUDIENCE: Yup.</span> </p>
<p><span m="2311930">PROFESSOR: All right, he</span> <span m="2312240">picked</span> <span m="2312440">6.</span> <span m="2312680">What</span> <span m="2312810">do</span> <span m="2312850">you</span> <span m="2312910">think</span> <span m="2313050">he</span> <span m="2313120">should</span> <span m="2313280">have done?</span> <span m="2313610">How</span> <span m="2313760">many</span> <span m="2313910">people</span> <span m="2314180">think</span> <span m="2314350">he</span> <span m="2314410">should</span> <span m="2314580">have</span> <span m="2314650">switched?</span> <span m="2316350">Ooh, Sean.</span> <span m="2317210">How</span> <span m="2317350">many</span> <span m="2317540">people</span> <span m="2317850">like</span> <span m="2318070">Sean's</span> <span m="2318430">choice?</span> <span m="2320010">Not</span> <span m="2320310">so</span> <span m="2320470">good.</span> <span m="2320700">All</span> <span m="2320810">right,</span> <span m="2320970">let's</span> <span m="2321130">see</span> <span m="2321240">if</span> <span m="2321350">you</span> <span m="2321450">won.</span> <span m="2325470">Here</span> <span m="2325740">we</span> <span m="2325870">go.</span> <span m="2326855">5.</span> <span m="2328550">Sean</span> <span m="2329110">wins</span> <span m="2329620">the</span> <span m="2329880">ice</span> <span m="2330150">cream.</span> <span m="2330560">Good</span> <span m="2330750">work.</span> <span m="2332150">Now</span> <span m="2332350">Sean,</span> <span m="2332680">what</span> <span m="2332840">was</span> <span m="2332970">your</span> <span m="2333110">thinking</span> <span m="2333530">here?</span> </p>
<p><span m="2336147">AUDIENCE: I was</span> <span m="2336634">thinking</span> <span m="2337121">[INAUDIBLE]</span> <span m="2338582">in the</span> <span m="2339069">other</span> <span m="2339556">envelope,</span> <span m="2340043">knowing</span> <span m="2340530">that</span> <span m="2341017">6</span> <span m="2342478">doesn't</span> <span m="2342965">tell me</span> <span m="2343452">anything about</span> <span m="2343939">what's in</span> <span m="2344913">the</span> <span m="2345400">other one,</span> <span m="2345887">so my</span> <span m="2346374">chances of</span> <span m="2346861">seeing</span> <span m="2347348">[INAUDIBLE].</span> </p>
<p><span m="2350760">PROFESSOR: Totally</span> <span m="2351290">50-50.</span> <span m="2352900">How</span> <span m="2353010">many</span> <span m="2353190">people</span> <span m="2353450">buy</span> <span m="2353700">that</span> <span m="2354025">argument?</span> <span m="2355130">It's</span> <span m="2355520">50-50.</span> <span m="2355845">He</span> <span m="2356170">really has</span> <span m="2356360">no idea</span> <span m="2356680">what's in</span> <span m="2357130">the other</span> <span m="2357360">envelope.</span> <span m="2357880">How</span> <span m="2358000">many</span> <span m="2358200">people</span> <span m="2358470">think</span> <span m="2358620">there's</span> <span m="2358770">a</span> <span m="2358820">better</span> <span m="2359040">way?</span> <span m="2361160">Not</span> <span m="2361400">too</span> <span m="2361540">many.</span> <span m="2362000">All</span> <span m="2362380">right, we're</span> <span m="2362640">going</span> <span m="2362780">to</span> <span m="2362830">try</span> <span m="2363050">it</span> <span m="2363130">again.</span> <span m="2363410">Well</span> <span m="2363670">done.</span> <span m="2364170">I've</span> <span m="2364460">got</span> <span m="2364660">different</span> <span m="2364950">envelopes</span> <span m="2365390">here.</span> <span m="2366310">And</span> <span m="2366530">tell</span> <span m="2366650">me</span> <span m="2366740">your</span> <span m="2366820">name</span> <span m="2367020">again?</span> </p>
<p><span m="2367470">AUDIENCE: Drew.</span> </p>
<p><span m="2367770">PROFESSOR: Drew,</span> <span m="2368070">OK</span> <span m="2368390">Drew.</span> <span m="2369510">Two</span> <span m="2369690">envelopes.</span> <span m="2371230">Which</span> <span m="2371410">one</span> <span m="2371480">would</span> <span m="2371640">you like</span> <span m="2371820">to</span> <span m="2371900">open?</span> </p>
<p><span m="2373136">AUDIENCE: They're both</span> <span m="2373594">labeled</span> <span m="2374052">B.</span> </p>
<p><span m="2374510">PROFESSOR: They're</span> <span m="2374640">both</span> <span m="2374920">labeled</span> <span m="2375240">B.</span> <span m="2375680">That won't</span> <span m="2375870">help</span> <span m="2376070">you</span> <span m="2376160">there.</span> <span m="2377460">All right,</span> <span m="2377880">he's</span> <span m="2378030">got</span> <span m="2378260">one.</span> <span m="2381600">Let's</span> <span m="2381780">see</span> <span m="2381930">what</span> <span m="2382060">you</span> <span m="2382170">got.</span> <span m="2389510">92.</span> <span m="2391620">Oh</span> <span m="2391950">my</span> <span m="2392180">goodness,</span> <span m="2392810">and</span> <span m="2392920">we've</span> <span m="2393080">seen</span> <span m="2393420">what</span> <span m="2393640">5</span> <span m="2393970">and</span> <span m="2394100">6.</span> <span m="2394520">What</span> <span m="2394680">are</span> <span m="2394710">you</span> <span m="2394790">going</span> <span m="2394880">to</span> <span m="2394980">do?</span> <span m="2395830">92.</span> <span m="2397300">What</span> <span m="2397620">should</span> <span m="2397750">he</span> <span m="2397930">do,</span> <span m="2398080">guys?</span> </p>
<p><span m="2398540">AUDIENCE: Stay.</span> </p>
<p><span m="2399180">PROFESSOR: Stay.</span> <span m="2399840">Oh</span> <span m="2400090">he's got</span> <span m="2400270">a</span> <span m="2400300">big</span> <span m="2400630">number</span> <span m="2400970">there.</span> <span m="2402780">You're going</span> <span m="2403050">to switch?</span> <span m="2404160">You're</span> <span m="2404270">giving</span> <span m="2404370">up a</span> <span m="2404470">92?</span> <span m="2406670">Ooh,</span> <span m="2407890">you're</span> <span m="2408090">sure?</span> <span m="2409480">You</span> <span m="2409600">don't</span> <span m="2409770">want</span> <span m="2409970">the</span> <span m="2410050">92?</span> <span m="2413140">How</span> <span m="2413250">many</span> <span m="2413440">people</span> <span m="2413760">think</span> <span m="2414040">Drew</span> <span m="2414320">is</span> <span m="2414540">going</span> <span m="2414650">to</span> <span m="2414720">win?</span> </p>
<p><span m="2415960">AUDIENCE: He's</span> <span m="2416100">trying to</span> <span m="2416170">lose.</span> </p>
<p><span m="2416890">PROFESSOR: A</span> <span m="2417180">couple</span> <span m="2417530">people.</span> <span m="2418250">Let's</span> <span m="2418500">see.</span> </p>
<p><span m="2420964">AUDIENCE: 91.</span> <span m="2424366">It's</span> <span m="2424852">either 91</span> <span m="2425338">or 93.</span> </p>
<p><span m="2426320">PROFESSOR: 93,</span> <span m="2427290">how</span> <span m="2427660">did you</span> <span m="2427790">know</span> <span m="2428030">there</span> <span m="2428160">was</span> <span m="2428310">a</span> <span m="2428380">bigger</span> <span m="2429000">number?</span> <span m="2429870">Wow,</span> <span m="2430570">very</span> <span m="2430910">good.</span> <span m="2431495">There you go.</span> <span m="2431770">What was</span> <span m="2432150">your</span> <span m="2432280">strategy?</span> </p>
<p><span m="2433500">AUDIENCE: I</span> <span m="2433970">figured</span> <span m="2434100">you'd</span> <span m="2434596">probably</span> <span m="2435092">pick</span> <span m="2435588">them within</span> <span m="2436084">1.</span> </p>
<p><span m="2436580">PROFESSOR: Yeah,</span> <span m="2437076">that's good.</span> </p>
<p><span m="2438068">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="2449972">PROFESSOR: Actually,</span> <span m="2450468">they're</span> <span m="2450964">both</span> <span m="2451460">[INAUDIBLE]</span> <span m="2451956">to 93.</span> <span m="2452452">Would</span> <span m="2452948">you still</span> <span m="2453444">switch?</span> </p>
<p><span m="2454350">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="2456560">PROFESSOR: Still</span> <span m="2456900">switch.</span> <span m="2457600">Hmm.</span> <span m="2458480">All right,</span> <span m="2458920">so</span> <span m="2459430">is</span> <span m="2459530">it</span> <span m="2459630">50-50</span> <span m="2460074">then?</span> <span m="2460518">Because you</span> <span m="2460962">see</span> <span m="2461406">a 92 a</span> <span m="2461850">93,</span> <span m="2462080">you're switching</span> <span m="2462520">either way.</span> </p>
<p><span m="2463410">AUDIENCE: Yeah.</span> </p>
<p><span m="2464300">PROFESSOR: 50-50.</span> <span m="2465635">All right,</span> <span m="2466080">how</span> <span m="2466150">many</span> <span m="2466320">people</span> <span m="2466610">still</span> <span m="2466820">like</span> <span m="2467030">50-50?</span> <span m="2467670">He can't</span> <span m="2468131">beat</span> <span m="2468592">50-50</span> <span m="2469053">here.</span> <span m="2470900">Any</span> <span m="2471060">ideas?</span> <span m="2471380">Can</span> <span m="2471470">you</span> <span m="2471890">beat</span> <span m="2472220">50-50?</span> <span m="2472840">Yeah.</span> </p>
<p><span m="2473796">AUDIENCE: I just</span> <span m="2474274">pay attention</span> <span m="2474752">to your face.</span> </p>
<p><span m="2476664">PROFESSOR: I</span> <span m="2477150">guess</span> <span m="2477310">I</span> <span m="2477380">gave</span> <span m="2477650">it</span> <span m="2477700">away.</span> <span m="2479391">I think</span> <span m="2479790">I've lost</span> <span m="2480050">every</span> <span m="2480270">bet</span> <span m="2480550">so</span> <span m="2480700">far</span> <span m="2480970">in</span> <span m="2481030">the</span> <span m="2481120">course</span> <span m="2481480">here,</span> <span m="2482300">even</span> <span m="2482500">on</span> <span m="2482550">Monty</span> <span m="2482930">Hall,</span> <span m="2483230">every</span> <span m="2483420">time.</span> <span m="2484320">So</span> <span m="2484430">I</span> <span m="2484520">guess</span> <span m="2485160">I</span> <span m="2485240">have</span> <span m="2485350">a</span> <span m="2485410">tell or</span> <span m="2485800">something</span> <span m="2486150">here.</span> <span m="2487450">Now,</span> <span m="2487670">in</span> <span m="2487810">fact,</span> <span m="2488390">you</span> <span m="2488560">can</span> <span m="2489280">beat</span> <span m="2489910">50-50.</span> <span m="2491210">The</span> <span m="2491330">information</span> <span m="2491910">is</span> <span m="2492130">helpful.</span> <span m="2492530">Now,</span> <span m="2493360">just</span> <span m="2493560">to</span> <span m="2493610">be</span> <span m="2493700">clear,</span> <span m="2494140">are</span> <span m="2494280">these</span> <span m="2494570">numbers</span> <span m="2494835">that</span> <span m="2495100">I</span> <span m="2495220">wrote</span> <span m="2495490">down</span> <span m="2495730">random?</span> <span m="2497760">No.</span> <span m="2498490">No,</span> <span m="2498740">I'm</span> <span m="2498940">trying</span> <span m="2499420">not</span> <span m="2499610">to</span> <span m="2499680">give</span> <span m="2499840">away--</span> <span m="2500100">my</span> <span m="2500300">ice</span> <span m="2500520">cream</span> <span m="2500710">bill is</span> <span m="2500930">going</span> <span m="2501090">through</span> <span m="2501150">the</span> <span m="2501260">roof</span> <span m="2501490">here.</span> <span m="2502180">I'm</span> <span m="2502340">trying</span> <span m="2502650">to</span> <span m="2502710">make</span> <span m="2502900">it</span> <span m="2502980">hard.</span> <span m="2503320">They're</span> <span m="2503700">very</span> <span m="2503950">much</span> <span m="2504210">not</span> <span m="2504790">random.</span> <span m="2505570">There</span> <span m="2505680">were</span> <span m="2505750">two</span> <span m="2505920">smalls</span> <span m="2506370">and</span> <span m="2506700">two</span> <span m="2506860">bigs.</span> <span m="2508400">What</span> <span m="2508640">is</span> <span m="2508870">random</span> <span m="2509280">here?</span> </p>
<p><span m="2511096">AUDIENCE: Which one</span> <span m="2511550">he picked</span> <span m="2512004">is.</span> </p>
<p><span m="2512460">PROFESSOR: Which</span> <span m="2512700">one</span> <span m="2512880">he</span> <span m="2512970">saw,</span> <span m="2513870">that's</span> <span m="2514050">50-50,</span> <span m="2515210">effectively.</span> <span m="2515810">So</span> <span m="2516290">that's</span> <span m="2516580">random</span> <span m="2516960">there.</span> <span m="2518920">Now</span> <span m="2520110">his</span> <span m="2520450">strategy</span> <span m="2521230">could</span> <span m="2521390">also</span> <span m="2521780">be</span> <span m="2521970">random,</span> <span m="2522560">but</span> <span m="2522600">it</span> <span m="2522720">wasn't.</span> <span m="2523810">His</span> <span m="2523990">strategy</span> <span m="2524450">was,</span> <span m="2524800">if</span> <span m="2524930">he sees</span> <span m="2525100">a</span> <span m="2525150">big</span> <span m="2525390">number,</span> <span m="2525710">he's</span> <span m="2525840">swapping,</span> <span m="2526780">which</span> <span m="2526940">is</span> <span m="2527050">odd.</span> <span m="2527320">Most</span> <span m="2527550">people</span> <span m="2527670">see</span> <span m="2527770">the</span> <span m="2527860">big</span> <span m="2528100">number,</span> <span m="2528310">they</span> <span m="2528420">want</span> <span m="2528580">to</span> <span m="2528630">keep</span> <span m="2528900">it.</span> <span m="2529980">You</span> <span m="2530070">know,</span> <span m="2530440">I'd have</span> <span m="2530540">done</span> <span m="2530900">well</span> <span m="2531110">today</span> <span m="2531330">if</span> <span m="2531450">I</span> <span m="2531630">had</span> <span m="2531780">a</span> <span m="2531870">really</span> <span m="2532110">big</span> <span m="2532310">number</span> <span m="2532570">and a</span> <span m="2532660">really</span> <span m="2532890">small</span> <span m="2533230">number,</span> <span m="2533460">because then</span> <span m="2534380">I</span> <span m="2534460">would have</span> <span m="2534660">won</span> <span m="2534920">both</span> <span m="2535190">times.</span> </p>
<p><span m="2539560">OK,</span> <span m="2540840">so</span> <span m="2541000">to</span> <span m="2541100">see</span> <span m="2541340">why</span> <span m="2541520">there</span> <span m="2541650">might</span> <span m="2542120">be</span> <span m="2542280">a</span> <span m="2542340">winning</span> <span m="2542630">strategy,</span> <span m="2543240">or</span> <span m="2543310">better</span> <span m="2543530">than</span> <span m="2543700">50-50,</span> <span m="2545020">imagine</span> <span m="2545590">that</span> <span m="2545730">I</span> <span m="2545830">had</span> <span m="2546900">been</span> <span m="2547120">nice</span> <span m="2547510">and</span> <span m="2547650">put</span> <span m="2547840">a</span> <span m="2547910">very</span> <span m="2548310">small</span> <span m="2548630">number</span> <span m="2548880">in</span> <span m="2548940">one</span> <span m="2549150">envelope</span> <span m="2550020">and</span> <span m="2550140">a</span> <span m="2550190">very</span> <span m="2550680">big</span> <span m="2551060">number</span> <span m="2551380">of</span> <span m="2551450">the</span> <span m="2551530">other</span> <span m="2551740">envelope.</span> <span m="2552640">Say</span> <span m="2552860">I</span> <span m="2552970">had</span> <span m="2553580">a</span> <span m="2553670">5</span> <span m="2554050">in</span> <span m="2554140">one</span> <span m="2554550">and</span> <span m="2554660">a</span> <span m="2554740">92</span> <span m="2555820">in</span> <span m="2555890">the</span> <span m="2556040">other.</span> <span m="2557520">Can</span> <span m="2557680">you</span> <span m="2557790">win</span> <span m="2558040">then,</span> <span m="2558580">if</span> <span m="2558700">you</span> <span m="2558800">know</span> <span m="2559060">that?</span> <span m="2561230">OK,</span> <span m="2561510">how</span> <span m="2561680">do</span> <span m="2561740">you</span> <span m="2561870">win</span> <span m="2562210">if</span> <span m="2562310">you</span> <span m="2562410">know</span> <span m="2562630">there's</span> <span m="2562810">one</span> <span m="2563020">less</span> <span m="2563280">than</span> <span m="2563420">10</span> <span m="2563670">and one</span> <span m="2563860">bigger</span> <span m="2564080">than</span> <span m="2564220">90?</span> <span m="2566100">Yeah?</span> </p>
<p><span m="2568141">AUDIENCE: If you get the one less than 10,</span> <span m="2568610">switch.</span> </p>
<p><span m="2569330">PROFESSOR: Yeah,</span> <span m="2569770">and</span> <span m="2570070">you're</span> <span m="2570160">going to</span> <span m="2570380">win with</span> <span m="2570530">what</span> <span m="2570730">probability?</span> <span m="2572820">1.</span> <span m="2573675">It's</span> <span m="2574050">a</span> <span m="2574160">certainty.</span> <span m="2575910">All</span> <span m="2575920">right,</span> <span m="2579230">well,</span> <span m="2580530">but</span> <span m="2580660">I'm</span> <span m="2580770">not</span> <span m="2580980">that</span> <span m="2581190">nice,</span> <span m="2581520">say.</span> <span m="2583700">Say,</span> <span m="2583980">though,</span> <span m="2584320">there</span> <span m="2584690">is</span> <span m="2584970">a</span> <span m="2585300">threshold</span> <span m="2586190">x,</span> <span m="2587600">such</span> <span m="2587880">that</span> <span m="2588060">one</span> <span m="2588420">of</span> <span m="2588470">the</span> <span m="2588540">numbers</span> <span m="2588940">is</span> <span m="2589080">less</span> <span m="2589440">than</span> <span m="2589580">x,</span> <span m="2590720">and</span> <span m="2590910">one</span> <span m="2591120">is</span> <span m="2591260">bigger</span> <span m="2591550">than</span> <span m="2591720">x,</span> <span m="2593030">and</span> <span m="2593220">you know</span> <span m="2593675">x.</span> <span m="2595510">Can</span> <span m="2595660">you</span> <span m="2595770">win</span> <span m="2595980">now?</span> <span m="2599600">Say</span> <span m="2599930">you</span> <span m="2600010">know</span> <span m="2600190">that</span> <span m="2600310">one</span> <span m="2600550">number is</span> <span m="2600900">less</span> <span m="2601290">than</span> <span m="2601460">47</span> <span m="2601890">and</span> <span m="2602320">1/2,</span> <span m="2602750">and</span> <span m="2602850">one</span> <span m="2603000">is</span> <span m="2603100">bigger</span> <span m="2603310">than</span> <span m="2603470">47</span> <span m="2604010">and</span> <span m="2604090">1/2.</span> <span m="2605700">Can</span> <span m="2605830">you</span> <span m="2605920">win?</span> <span m="2607380">Yeah,</span> <span m="2608540">because</span> <span m="2608860">if</span> <span m="2608950">you</span> <span m="2609040">get</span> <span m="2609210">the</span> <span m="2609290">one</span> <span m="2609470">less</span> <span m="2609760">than</span> <span m="2609890">47</span> <span m="2610210">and</span> <span m="2610530">1/2,</span> <span m="2611030">you</span> <span m="2611180">switch.</span> <span m="2612070">And</span> <span m="2612390">otherwise</span> <span m="2612760">you'll stay.</span> <span m="2613030">So</span> <span m="2613300">again,</span> <span m="2613620">you</span> <span m="2613750">win</span> <span m="2614030">with</span> <span m="2614160">certainty.</span> <span m="2616240">All</span> <span m="2616400">right,</span> <span m="2616560">that's</span> <span m="2616820">good.</span> </p>
<p><span m="2618730">Is</span> <span m="2618960">there</span> <span m="2619110">always</span> <span m="2619530">such</span> <span m="2619780">an</span> <span m="2619870">x?</span> <span m="2623400">You</span> <span m="2623520">may</span> <span m="2623650">not</span> <span m="2623940">know</span> <span m="2624150">it,</span> <span m="2624380">but</span> <span m="2624450">is</span> <span m="2624690">there</span> <span m="2624830">always</span> <span m="2625180">such</span> <span m="2625400">an</span> <span m="2625490">x?</span> <span m="2626810">Yeah,</span> <span m="2627200">because</span> <span m="2627350">the</span> <span m="2627440">numbers</span> <span m="2627720">are</span> <span m="2627780">different.</span> <span m="2628840">Just</span> <span m="2629050">pick</span> <span m="2629220">up</span> <span m="2629480">a</span> <span m="2629990">real</span> <span m="2630200">number in</span> <span m="2630480">between</span> <span m="2630820">them.</span> <span m="2631080">You know,</span> <span m="2631520">with</span> <span m="2631820">92 and</span> <span m="2632250">93,</span> <span m="2632940">there's</span> <span m="2633200">92</span> <span m="2633490">and</span> <span m="2633790">1/2,</span> <span m="2634110">is</span> <span m="2634330">such</span> <span m="2634580">an</span> <span m="2634660">x.</span> <span m="2635670">Now</span> <span m="2635780">the</span> <span m="2635930">only</span> <span m="2636140">problem</span> <span m="2636600">is</span> <span m="2637230">you</span> <span m="2637380">don't</span> <span m="2637600">know</span> <span m="2637810">it.</span> <span m="2639470">And</span> <span m="2639650">there's</span> <span m="2639840">no</span> <span m="2640160">way</span> <span m="2641010">to</span> <span m="2641110">figure</span> <span m="2641400">it</span> <span m="2641460">out</span> <span m="2641710">that's</span> <span m="2641920">within</span> <span m="2642170">the</span> <span m="2642260">rules</span> <span m="2642490">of</span> <span m="2642560">the</span> <span m="2642610">game.</span> </p>
<p><span m="2644580">Now</span> <span m="2645130">in</span> <span m="2645290">life,</span> <span m="2646570">if</span> <span m="2646710">you</span> <span m="2646830">don't</span> <span m="2647080">know</span> <span m="2647300">something,</span> <span m="2648970">and</span> <span m="2649120">you</span> <span m="2649230">can't</span> <span m="2649680">figure it</span> <span m="2650050">out,</span> <span m="2651290">what</span> <span m="2651510">can</span> <span m="2651730">you</span> <span m="2651830">do?</span> </p>
<p><span m="2652870">AUDIENCE: Guess.</span> </p>
<p><span m="2653670">PROFESSOR: Guess</span> <span m="2654080">it.</span> <span m="2655340">All right,</span> <span m="2655530">now</span> <span m="2655640">this</span> <span m="2655790">turns</span> <span m="2656010">out</span> <span m="2656120">to</span> <span m="2656160">be</span> <span m="2656390">a</span> <span m="2656490">really</span> <span m="2656810">good</span> <span m="2657070">thing</span> <span m="2657250">to</span> <span m="2657310">do,</span> <span m="2658080">and</span> <span m="2658190">especially</span> <span m="2658710">good</span> <span m="2658840">in</span> <span m="2658900">a</span> <span m="2658950">lot</span> <span m="2659220">of</span> <span m="2659280">computer</span> <span m="2659670">science</span> <span m="2660450">situations.</span> <span m="2661890">If</span> <span m="2662010">you</span> <span m="2662110">don't</span> <span m="2662410">know</span> <span m="2662620">it,</span> <span m="2662690">and</span> <span m="2662780">you</span> <span m="2662850">can't</span> <span m="2663190">know</span> <span m="2663310">it,</span> <span m="2663590">well you</span> <span m="2663890">could</span> <span m="2664010">try</span> <span m="2664160">to</span> <span m="2664240">guess</span> <span m="2665200">and</span> <span m="2665500">you</span> <span m="2665610">might</span> <span m="2666030">be</span> <span m="2666170">right.</span> <span m="2668210">Now,</span> <span m="2668390">if</span> <span m="2668530">you</span> <span m="2668620">guess</span> <span m="2669010">x</span> <span m="2669270">and</span> <span m="2669450">you</span> <span m="2669570">are</span> <span m="2669730">right,</span> <span m="2671570">what's</span> <span m="2671810">your</span> <span m="2671920">probability</span> <span m="2672400">of</span> <span m="2672470">winning?</span> <span m="2674060">1.</span> <span m="2675970">If</span> <span m="2676130">you</span> <span m="2676290">don't</span> <span m="2676670">guess</span> <span m="2677060">x,</span> <span m="2679000">what's</span> <span m="2679200">your</span> <span m="2679350">probability</span> <span m="2679900">of</span> <span m="2679960">winning?</span> <span m="2680220">Say</span> <span m="2680350">you</span> <span m="2680510">guess</span> <span m="2680940">wrong,</span> <span m="2682850">but</span> <span m="2683000">you</span> <span m="2683090">follow</span> <span m="2683520">the</span> <span m="2683670">rule</span> <span m="2684080">that if</span> <span m="2684180">it's</span> <span m="2684340">less</span> <span m="2684750">than</span> <span m="2684920">what</span> <span m="2685090">you</span> <span m="2685170">guessed,</span> <span m="2685550">you</span> <span m="2685640">swap</span> <span m="2686030">it,</span> <span m="2686100">otherwise</span> <span m="2686420">you</span> <span m="2686500">don't.</span> <span m="2687560">What's</span> <span m="2687750">your</span> <span m="2687870">probability</span> <span m="2688340">of</span> <span m="2688410">winning</span> <span m="2688740">that?</span> <span m="2691370">P?</span> <span m="2691960">Well,</span> <span m="2692340">yeah,</span> <span m="2692590">what's</span> <span m="2692790">P?</span> <span m="2695490">It's</span> <span m="2695630">a</span> <span m="2695690">nice</span> <span m="2696030">value</span> <span m="2696350">of P.</span> </p>
<p><span m="2698840">You know,</span> <span m="2698980">say</span> <span m="2699200">you</span> <span m="2699270">guessed--</span> <span m="2700470">well,</span> <span m="2700900">say</span> <span m="2701020">you</span> <span m="2701450">weren't</span> <span m="2701620">paying</span> <span m="2701810">attention and</span> <span m="2702310">you</span> <span m="2702320">guessed</span> <span m="2702640">200.</span> <span m="2705080">And</span> <span m="2705140">your</span> <span m="2705320">rule</span> <span m="2705640">is,</span> <span m="2705800">if</span> <span m="2705910">you're</span> <span m="2706050">less</span> <span m="2706370">than</span> <span m="2706460">what</span> <span m="2706610">you</span> <span m="2706690">guessed,</span> <span m="2707010">you're</span> <span m="2707130">less than</span> <span m="2707450">200,</span> <span m="2707750">you</span> <span m="2707900">swap,</span> <span m="2709090">which</span> <span m="2709320">means</span> <span m="2709650">you're</span> <span m="2709800">just</span> <span m="2710020">going</span> <span m="2710110">to</span> <span m="2710160">swap.</span> <span m="2712040">And</span> <span m="2712260">you</span> <span m="2712340">started</span> <span m="2712720">with</span> <span m="2712850">a</span> <span m="2712910">random</span> <span m="2713270">one,</span> <span m="2714400">so</span> <span m="2714540">what's</span> <span m="2714690">your</span> <span m="2714790">chance</span> <span m="2715060">of</span> <span m="2715140">winning?</span> <span m="2717370">A half.</span> <span m="2719190">So</span> <span m="2719300">if you</span> <span m="2719430">guessed</span> <span m="2719820">wrong,</span> <span m="2720430">you didn't</span> <span m="2720740">lose</span> <span m="2721180">anything.</span> <span m="2721670">You</span> <span m="2721770">still</span> <span m="2721940">win</span> <span m="2722100">with</span> <span m="2722230">probability</span> <span m="2722710">a half.</span> <span m="2723580">If</span> <span m="2723750">you</span> <span m="2723840">guessed</span> <span m="2724170">right,</span> <span m="2725560">you</span> <span m="2725700">win</span> <span m="2725900">with</span> <span m="2726030">probability</span> <span m="2726540">1,</span> <span m="2727900">and</span> <span m="2728050">there's</span> <span m="2728200">some</span> <span m="2728470">chance</span> <span m="2728790">you</span> <span m="2728860">guessed</span> <span m="2729130">right,</span> <span m="2729610">so</span> <span m="2729810">now</span> <span m="2730140">you've</span> <span m="2730290">got a</span> <span m="2730490">strategy</span> <span m="2731000">that</span> <span m="2731110">beats</span> <span m="2731370">50-50,</span> <span m="2733250">because</span> <span m="2733670">you</span> <span m="2733760">guessed.</span> </p>
<p><span m="2735970">OK, and</span> <span m="2736210">this</span> <span m="2736670">is</span> <span m="2736810">a</span> <span m="2736860">whole</span> <span m="2737230">field</span> <span m="2737680">in</span> <span m="2738100">computer</span> <span m="2738490">science</span> <span m="2739040">where</span> <span m="2739470">you</span> <span m="2739560">get</span> <span m="2739810">randomized</span> <span m="2740640">algorithms,</span> <span m="2741580">where</span> <span m="2741840">this</span> <span m="2742040">strategy</span> <span m="2742900">depends</span> <span m="2743360">on</span> <span m="2743490">a</span> <span m="2743540">coin</span> <span m="2743860">flip,</span> <span m="2744390">on</span> <span m="2744640">guessing</span> <span m="2745130">a</span> <span m="2745150">value.</span> <span m="2746760">And</span> <span m="2746850">it</span> <span m="2746950">leads</span> <span m="2747300">to</span> <span m="2747420">getting</span> <span m="2747890">potentially</span> <span m="2748990">a</span> <span m="2749070">better</span> <span m="2750060">outcome.</span> <span m="2751360">All right,</span> <span m="2751760">so</span> <span m="2751860">let's</span> <span m="2752300">prove</span> <span m="2752770">that</span> <span m="2753030">now</span> <span m="2753260">and</span> <span m="2753380">see</span> <span m="2753680">what</span> <span m="2754000">the</span> <span m="2754100">probability</span> <span m="2754820">is</span> <span m="2755560">of</span> <span m="2755700">winning</span> <span m="2756570">with</span> <span m="2756840">the</span> <span m="2756940">guess</span> <span m="2757320">strategy</span> <span m="2757850">and</span> <span m="2757920">what's</span> <span m="2758140">a</span> <span m="2758200">good</span> <span m="2758360">way</span> <span m="2758510">to</span> <span m="2758980">guess.</span> </p>
<p><span m="2767290">So</span> <span m="2767450">we're</span> <span m="2767550">going</span> <span m="2767630">to</span> <span m="2767690">first</span> <span m="2768300">formalize</span> <span m="2769030">our</span> <span m="2769920">random</span> <span m="2770440">protocol.</span> <span m="2781350">All right,</span> <span m="2781550">so if</span> <span m="2781840">this</span> <span m="2782040">is</span> <span m="2782170">the</span> <span m="2782290">winning</span> <span m="2782530">strategy,</span> <span m="2790500">well</span> <span m="2790770">first</span> <span m="2793370">the</span> <span m="2793490">envelopes</span> <span m="2796220">contain</span> <span m="2799190">y</span> <span m="2800050">and</span> <span m="2800250">z,</span> <span m="2802020">and they're</span> <span m="2802385">in</span> <span m="2802750">the</span> <span m="2802880">interval</span> <span m="2803240">0</span> <span m="2803590">to</span> <span m="2803700">n.</span> <span m="2805730">And</span> <span m="2806250">y</span> <span m="2806570">is</span> <span m="2806710">going</span> <span m="2806820">to</span> <span m="2806860">be</span> <span m="2806960">less</span> <span m="2807240">than</span> <span m="2807380">z.</span> <span m="2807720">That's</span> <span m="2807920">how</span> <span m="2808000">we're</span> <span m="2808110">going</span> <span m="2808250">to</span> <span m="2808320">define</span> <span m="2808580">them.</span> <span m="2809170">Now</span> <span m="2809370">you</span> <span m="2809520">don't</span> <span m="2809880">know</span> <span m="2810820">y</span> <span m="2811130">and</span> <span m="2811280">z,</span> <span m="2811660">but</span> <span m="2812920">y</span> <span m="2813290">is</span> <span m="2813450">the</span> <span m="2813580">smaller</span> <span m="2814160">number</span> <span m="2814430">in the</span> <span m="2814550">envelope,</span> <span m="2814910">z</span> <span m="2815130">is</span> <span m="2815270">the</span> <span m="2815330">larger.</span> <span m="2816730">And</span> <span m="2817100">in</span> <span m="2817190">our</span> <span m="2817340">example,</span> <span m="2817820">n</span> <span m="2818030">was</span> <span m="2818200">100.</span> </p>
<p><span m="2821230">Now</span> <span m="2821660">the</span> <span m="2821750">player</span> <span m="2825560">chooses</span> <span m="2829380">x</span> <span m="2831390">randomly,</span> <span m="2832540">uniformly</span> <span m="2835930">among</span> <span m="2836270">all</span> <span m="2836660">the</span> <span m="2836760">possible</span> <span m="2838080">half</span> <span m="2838420">integers.</span> <span m="2841520">So</span> <span m="2841700">he</span> <span m="2841810">might</span> <span m="2842020">pick</span> <span m="2842180">1/2,</span> <span m="2842680">he</span> <span m="2842760">might</span> <span m="2842960">pick</span> <span m="2843150">1 and</span> <span m="2843420">1/2,</span> <span m="2844890">2</span> <span m="2845060">and</span> <span m="2845170">1/2,</span> <span m="2846230">all</span> <span m="2846530">the</span> <span m="2846610">way</span> <span m="2846830">out</span> <span m="2847080">to</span> <span m="2847690">n</span> <span m="2847910">minus</span> <span m="2848320">1/2.</span> <span m="2850001">You know,</span> <span m="2850450">because</span> <span m="2850710">if the</span> <span m="2850790">player</span> <span m="2851110">picks</span> <span m="2851360">anything</span> <span m="2851710">else,</span> <span m="2852010">it's</span> <span m="2852170">useless.</span> <span m="2853500">He</span> <span m="2853630">won't</span> <span m="2853830">have</span> <span m="2853980">a</span> <span m="2854030">chance</span> <span m="2854330">to</span> <span m="2854450">win.</span> <span m="2855590">But</span> <span m="2855730">he</span> <span m="2855880">might,</span> <span m="2856380">in</span> <span m="2856610">the</span> <span m="2856690">case</span> <span m="2857040">of</span> <span m="2857170">5 and</span> <span m="2857560">6,</span> <span m="2857880">might</span> <span m="2858140">have</span> <span m="2858260">picked</span> <span m="2858580">5</span> <span m="2858850">and</span> <span m="2859110">1/2.</span> </p>
<p><span m="2860570">And</span> <span m="2860770">we're</span> <span m="2860890">going</span> <span m="2860980">to</span> <span m="2861530">make</span> <span m="2861920">our</span> <span m="2862160">random</span> <span m="2862550">guess</span> <span m="2863020">be</span> <span m="2863200">equally</span> <span m="2863650">likely</span> <span m="2864100">among</span> <span m="2864400">those</span> <span m="2864750">n</span> <span m="2865310">values,</span> <span m="2865840">because</span> <span m="2867420">well he</span> <span m="2867550">doesn't</span> <span m="2867800">really</span> <span m="2868060">know</span> <span m="2869630">what</span> <span m="2870540">the</span> <span m="2870740">numbers</span> <span m="2871070">are</span> <span m="2871280">I put</span> <span m="2871470">in</span> <span m="2871590">the</span> <span m="2871850">envelopes.</span> <span m="2872290">If</span> <span m="2872310">he</span> <span m="2872400">sees</span> <span m="2872610">6, he</span> <span m="2873010">doesn't</span> <span m="2873210">know</span> <span m="2873330">if</span> <span m="2873400">it's</span> <span m="2873560">5</span> <span m="2873940">or it's</span> <span m="2874120">7,</span> <span m="2875730">OK?</span> <span m="2877130">So</span> <span m="2877250">that's</span> <span m="2877480">why</span> <span m="2877630">we're</span> <span m="2877740">going</span> <span m="2877820">to</span> <span m="2877880">pick</span> <span m="2878050">them</span> <span m="2878190">all</span> <span m="2878420">equally</span> <span m="2878720">likely</span> <span m="2879350">and it's</span> <span m="2879550">uniform.</span> </p>
<p><span m="2882860">Now</span> <span m="2883060">the</span> <span m="2883170">player</span> <span m="2883670">is</span> <span m="2883780">hoping</span> <span m="2886600">that</span> <span m="2886750">he</span> <span m="2886920">picked</span> <span m="2887190">a</span> <span m="2887270">value</span> <span m="2887700">that</span> <span m="2887850">splits</span> <span m="2888570">y</span> <span m="2889010">and</span> <span m="2889160">z,</span> <span m="2892170">OK?</span> <span m="2892500">Because</span> <span m="2892830">then</span> <span m="2893160">he's</span> <span m="2893340">going</span> <span m="2893450">to</span> <span m="2893530">win.</span> <span m="2895460">If he</span> <span m="2895480">picked an</span> <span m="2895690">x</span> <span m="2895990">between</span> <span m="2896460">the</span> <span m="2896530">numbers</span> <span m="2896840">in</span> <span m="2896880">the</span> <span m="2897020">envelopes,</span> <span m="2897620">and</span> <span m="2897750">follows</span> <span m="2898170">the</span> <span m="2898300">swap,</span> <span m="2899180">if</span> <span m="2899370">he</span> <span m="2899480">got</span> <span m="2899650">the</span> <span m="2899730">small</span> <span m="2900170">number</span> <span m="2900550">less</span> <span m="2900770">than</span> <span m="2901020">x,</span> <span m="2901860">he's</span> <span m="2902020">going</span> <span m="2902130">to</span> <span m="2902210">win,</span> <span m="2903968">all</span> <span m="2904444">right?</span> <span m="2907050">Then</span> <span m="2907240">the</span> <span m="2907340">player</span> <span m="2908160">opens</span> <span m="2908570">a</span> <span m="2908660">random</span> <span m="2909030">envelope,</span> <span m="2915310">50-50,</span> <span m="2921100">to</span> <span m="2921300">reveal</span> <span m="2922140">a</span> <span m="2922470">number</span> <span m="2923100">r,</span> <span m="2925020">which</span> <span m="2925240">is</span> <span m="2925470">either</span> <span m="2926130">y</span> <span m="2926530">or</span> <span m="2926700">z,</span> <span m="2927340">but</span> <span m="2927740">the</span> <span m="2927840">player</span> <span m="2928190">doesn't</span> <span m="2928510">know</span> <span m="2928830">which</span> <span m="2929060">one.</span> <span m="2931730">So</span> <span m="2931930">one</span> <span m="2932240">of</span> <span m="2932290">the</span> <span m="2932380">numbers</span> <span m="2932740">gets</span> <span m="2932970">revealed.</span> <span m="2935690">And</span> <span m="2935840">then</span> <span m="2935980">the</span> <span m="2936070">last</span> <span m="2936410">step</span> <span m="2936640">is</span> <span m="2938180">the</span> <span m="2938260">player</span> <span m="2938740">swaps</span> <span m="2943100">if</span> <span m="2943460">the</span> <span m="2943570">number</span> <span m="2944000">he</span> <span m="2944190">revealed</span> <span m="2945330">is</span> <span m="2945660">less</span> <span m="2946090">than</span> <span m="2947400">the</span> <span m="2947520">guessed</span> <span m="2948400">split</span> <span m="2948850">number.</span> </p>
<p><span m="2952580">That's</span> <span m="2952940">the</span> <span m="2953310">strategy,</span> <span m="2955550">and</span> <span m="2955710">it's</span> <span m="2955840">a</span> <span m="2955900">strategy</span> <span m="2956400">that</span> <span m="2956520">depends</span> <span m="2956795">on</span> <span m="2958880">a</span> <span m="2959030">random</span> <span m="2959480">guess</span> <span m="2961980">or</span> <span m="2962120">a</span> <span m="2962160">random</span> <span m="2962500">number.</span> <span m="2966410">So</span> <span m="2966470">let's</span> <span m="2966690">figure</span> <span m="2966970">out</span> <span m="2967110">the</span> <span m="2967190">probability</span> <span m="2967760">of</span> <span m="2967840">winning</span> <span m="2968370">with</span> <span m="2968480">that</span> <span m="2968660">strategy.</span> <span m="2970060">And</span> <span m="2970180">we're</span> <span m="2970280">going</span> <span m="2970370">to</span> <span m="2970410">use</span> <span m="2970700">the</span> <span m="2970780">tree</span> <span m="2970980">method.</span> <span m="2973640">Well,</span> <span m="2975050">the</span> <span m="2975150">first</span> <span m="2976230">branch</span> <span m="2976670">in</span> <span m="2976710">the</span> <span m="2976790">tree</span> <span m="2977040">method</span> <span m="2977410">is</span> <span m="2978810">whether</span> <span m="2979040">or</span> <span m="2979060">not--</span> <span m="2979400">where</span> <span m="2979660">the</span> <span m="2979760">guess</span> <span m="2980120">wound</span> <span m="2980460">up.</span> <span m="2980700">And</span> <span m="2980830">there's</span> <span m="2981020">three</span> <span m="2981240">cases</span> <span m="2981720">for</span> <span m="2981910">how</span> <span m="2982120">the</span> <span m="2982210">guess</span> <span m="2982610">did.</span> </p>
<p><span m="2985320">You</span> <span m="2985460">might</span> <span m="2985690">have</span> <span m="2985850">guessed</span> <span m="2986350">too</span> <span m="2986550">low,</span> <span m="2987930">in which</span> <span m="2988320">case,</span> <span m="2988660">x</span> <span m="2989040">is</span> <span m="2989210">less</span> <span m="2989480">than</span> <span m="2989650">y,</span> <span m="2991280">and</span> <span m="2991590">y</span> <span m="2991820">is</span> <span m="2991920">less</span> <span m="2992110">than</span> <span m="2992240">z.</span> <span m="2993590">You</span> <span m="2993730">might have</span> <span m="2993960">guessed</span> <span m="2994500">perfectly,</span> <span m="2995770">in</span> <span m="2995920">which</span> <span m="2996190">case</span> <span m="2997130">x</span> <span m="2997430">is</span> <span m="2997630">between</span> <span m="2998470">y</span> <span m="2998730">and</span> <span m="2998870">z,</span> <span m="3000540">equals</span> <span m="3000910">not</span> <span m="3001120">possible.</span> <span m="3002570">Or</span> <span m="3002860">you</span> <span m="3002970">might</span> <span m="3003260">have guessed</span> <span m="3003580">too</span> <span m="3003810">high,</span> <span m="3004840">in</span> <span m="3004890">which</span> <span m="3005060">case</span> <span m="3005330">y</span> <span m="3005650">is</span> <span m="3005780">less</span> <span m="3005990">than</span> <span m="3006130">z</span> <span m="3006370">is</span> <span m="3006530">less</span> <span m="3006770">than</span> <span m="3006860">x.</span> <span m="3008470">All</span> <span m="3008550">right,</span> <span m="3008730">so</span> <span m="3008930">here's</span> <span m="3009360">low,</span> <span m="3010820">here's</span> <span m="3011070">high,</span> <span m="3012430">and</span> <span m="3012540">here's</span> <span m="3013545">an</span> <span m="3013990">OK</span> <span m="3014360">guess,</span> <span m="3014615">a</span> <span m="3014870">good</span> <span m="3015060">guess.</span> </p>
<p><span m="3017270">Now</span> <span m="3017460">what's</span> <span m="3017810">the</span> <span m="3017920">probability</span> <span m="3018680">you</span> <span m="3018800">guessed</span> <span m="3019530">low,</span> <span m="3021335">that</span> <span m="3021720">you</span> <span m="3021920">picked</span> <span m="3022260">an</span> <span m="3022390">x</span> <span m="3022780">less</span> <span m="3023100">than</span> <span m="3023240">y?</span> <span m="3025130">y</span> <span m="3025410">is</span> <span m="3025510">integers,</span> <span m="3026870">x</span> <span m="3027220">is</span> <span m="3027360">the</span> <span m="3027440">half</span> <span m="3027820">integers.</span> <span m="3030140">What's</span> <span m="3030450">the</span> <span m="3030540">probability</span> <span m="3031250">to</span> <span m="3031340">assign</span> <span m="3031810">to</span> <span m="3031900">that</span> <span m="3033240">chance,</span> <span m="3033690">that you</span> <span m="3033820">guessed</span> <span m="3034200">low?</span> <span m="3036925">You could use</span> <span m="3037392">the</span> <span m="3037859">value of</span> <span m="3038330">y in</span> <span m="3038670">your</span> <span m="3038940">answer.</span> <span m="3039560">You</span> <span m="3039670">don't</span> <span m="3039860">know</span> <span m="3040060">y,</span> <span m="3040350">but</span> <span m="3040550">we</span> <span m="3040630">can</span> <span m="3040840">write</span> <span m="3041030">it</span> <span m="3041090">on</span> <span m="3041190">the</span> <span m="3041270">tree.</span> <span m="3044790">What</span> <span m="3044950">if</span> <span m="3045090">y</span> <span m="3045490">was</span> <span m="3045710">1?</span> <span m="3047730">The</span> <span m="3047830">smallest</span> <span m="3048430">number</span> <span m="3048690">in the</span> <span m="3049100">envelopes is</span> <span m="3049270">one,</span> <span m="3049970">what's</span> <span m="3050330">the</span> <span m="3050430">probability</span> <span m="3051120">you</span> <span m="3051280">guessed</span> <span m="3052800">below</span> <span m="3053850">1?</span> </p>
<p><span m="3057660">Not</span> <span m="3057850">quite</span> <span m="3058080">0,</span> <span m="3060250">one</span> <span m="3060620">of</span> <span m="3060710">your</span> <span m="3060910">possible</span> <span m="3062080">end</span> <span m="3062400">guesses</span> <span m="3062950">would</span> <span m="3063640">be</span> <span m="3063860">too</span> <span m="3064010">low,</span> <span m="3064650">namely,</span> <span m="3064970">a half.</span> <span m="3065920">If</span> <span m="3067480">the</span> <span m="3067600">smallest</span> <span m="3068180">number</span> <span m="3068460">y</span> <span m="3068810">were</span> <span m="3069030">2,</span> <span m="3069950">what's</span> <span m="3070170">the</span> <span m="3070250">probability</span> <span m="3070740">you guess</span> <span m="3071280">below</span> <span m="3071730">2?</span> <span m="3073340">2/n.</span> <span m="3075070">And</span> <span m="3075250">in</span> <span m="3075380">general,</span> <span m="3076090">if</span> <span m="3076330">y</span> <span m="3076940">is</span> <span m="3077810">the</span> <span m="3077920">smallest</span> <span m="3078400">value,</span> <span m="3079510">you've</span> <span m="3079690">got</span> <span m="3079900">y</span> <span m="3080390">possible</span> <span m="3080920">guesses</span> <span m="3081350">that it</span> <span m="3081440">would</span> <span m="3081550">be</span> <span m="3081670">too</span> <span m="3081850">low,</span> <span m="3083070">namely</span> <span m="3083330">all</span> <span m="3083510">the</span> <span m="3083580">half</span> <span m="3083880">integers</span> <span m="3084280">less</span> <span m="3084500">than</span> <span m="3084610">y.</span> <span m="3087070">So</span> <span m="3087190">the</span> <span m="3087270">probability</span> <span m="3088010">you</span> <span m="3088110">guess</span> <span m="3088410">low</span> <span m="3088990">is</span> <span m="3089140">y/n,</span> <span m="3091920">because</span> <span m="3092080">each</span> <span m="3092800">one</span> <span m="3093060">is</span> <span m="3093230">1</span> <span m="3093580">in n</span> <span m="3093710">chance.</span> <span m="3094190">And there's</span> <span m="3094420">y</span> <span m="3094850">that</span> <span m="3094980">are</span> <span m="3095050">too</span> <span m="3095220">low.</span> </p>
<p><span m="3097250">What's</span> <span m="3097500">the</span> <span m="3097600">probability</span> <span m="3098160">your</span> <span m="3098250">guess is</span> <span m="3098670">good,</span> <span m="3099070">that you</span> <span m="3099150">split</span> <span m="3099540">y</span> <span m="3099810">and</span> <span m="3099930">z?</span> <span m="3104930">z</span> <span m="3105356">minus</span> <span m="3106210">y</span> <span m="3108440">over</span> <span m="3108790">n,</span> <span m="3110530">because</span> <span m="3110930">they're--</span> <span m="3111730">between</span> <span m="3112120">these</span> <span m="3112310">integers,</span> <span m="3112690">z</span> <span m="3112870">and</span> <span m="3112960">y,</span> <span m="3113260">there's</span> <span m="3113410">z</span> <span m="3113710">minus</span> <span m="3114100">y</span> <span m="3114850">half</span> <span m="3115150">integers.</span> <span m="3116910">Each</span> <span m="3117180">could</span> <span m="3117290">have</span> <span m="3117380">been</span> <span m="3117540">guessed</span> <span m="3117810">with</span> <span m="3117920">probability</span> <span m="3118330">1/n.</span> <span m="3119990">And</span> <span m="3120220">what's</span> <span m="3120420">the</span> <span m="3120510">probability</span> <span m="3120780">you</span> <span m="3121050">guessed</span> <span m="3121350">high?</span> <span m="3124330">n</span> <span m="3124580">minus</span> <span m="3124970">z</span> <span m="3125800">over</span> <span m="3126130">n,</span> <span m="3126940">because</span> <span m="3127330">there's</span> <span m="3127560">n</span> <span m="3127770">minus</span> <span m="3128130">g</span> <span m="3128410">half</span> <span m="3128740">integers</span> <span m="3130360">between</span> <span m="3130990">z</span> <span m="3131856">and</span> <span m="3132620">n</span> <span m="3132970">being</span> <span m="3133080">the</span> <span m="3133170">last</span> <span m="3133530">possible</span> <span m="3133930">one,</span> <span m="3134930">and</span> <span m="3135080">minus the</span> <span m="3135370">half</span> <span m="3135660">there.</span> </p>
<p><span m="3136570">All</span> <span m="3136730">right,</span> <span m="3137010">so</span> <span m="3137110">we've</span> <span m="3137260">got</span> <span m="3138200">the</span> <span m="3138300">first</span> <span m="3138640">branch.</span> <span m="3139620">Now</span> <span m="3139790">the</span> <span m="3139900">next</span> <span m="3141750">branch</span> <span m="3142760">is</span> <span m="3143030">the</span> <span m="3143150">revealed</span> <span m="3143670">value.</span> <span m="3144620">Did</span> <span m="3144830">you</span> <span m="3145280">open</span> <span m="3145820">the</span> <span m="3145910">smaller</span> <span m="3146440">one</span> <span m="3147130">or</span> <span m="3147260">the</span> <span m="3147390">larger</span> <span m="3147790">one?</span> <span m="3149190">r</span> <span m="3149450">equals</span> <span m="3149750">y</span> <span m="3150120">means</span> <span m="3150360">you</span> <span m="3150490">opened</span> <span m="3150690">the</span> <span m="3150750">smaller</span> <span m="3151180">one.</span> <span m="3151430">r</span> <span m="3151720">equals</span> <span m="3152010">z</span> <span m="3152260">means</span> <span m="3152480">you</span> <span m="3152590">saw</span> <span m="3152780">the</span> <span m="3152900">bigger</span> <span m="3153180">one.</span> </p>
<p><span m="3162580">All right,</span> <span m="3162925">if</span> <span m="3163270">I've</span> <span m="3163610">gone</span> <span m="3163920">down</span> <span m="3164150">this</span> <span m="3164350">branch</span> <span m="3164800">where I</span> <span m="3165190">guessed</span> <span m="3165520">too</span> <span m="3165710">low,</span> <span m="3166090">but</span> <span m="3166660">I</span> <span m="3166780">don't</span> <span m="3166960">know</span> <span m="3167150">that,</span> <span m="3167380">yet,</span> <span m="3167750">but</span> <span m="3167930">say</span> <span m="3168100">I've</span> <span m="3168640">guessed</span> <span m="3168860">too</span> <span m="3169050">low,</span> <span m="3169610">what's</span> <span m="3169880">the</span> <span m="3169970">probability</span> <span m="3170540">I</span> <span m="3170720">opened</span> <span m="3171170">the</span> <span m="3171270">smaller</span> <span m="3171850">envelope?</span> <span m="3175830">One half,</span> <span m="3176600">because</span> <span m="3177280">you're</span> <span m="3177360">just</span> <span m="3177550">picking</span> <span m="3177720">a</span> <span m="3177850">random</span> <span m="3178130">envelope</span> <span m="3178500">and</span> <span m="3178590">opening</span> <span m="3178950">up.</span> <span m="3179870">So</span> <span m="3179980">these</span> <span m="3180230">each</span> <span m="3180440">happen</span> <span m="3180780">with</span> <span m="3180920">probability</span> <span m="3181360">a half.</span> <span m="3182790">And</span> <span m="3182880">that's</span> <span m="3183450">true</span> <span m="3183650">no</span> <span m="3183730">matter</span> <span m="3183970">what.</span> <span m="3185390">They're</span> <span m="3185520">all</span> <span m="3185740">one half.</span> </p>
<p><span m="3188530">All</span> <span m="3188760">right,</span> <span m="3191770">well,</span> <span m="3191910">now</span> <span m="3192110">we</span> <span m="3192230">can</span> <span m="3192360">compute</span> <span m="3192690">the</span> <span m="3192780">probability</span> <span m="3193370">of</span> <span m="3193440">each</span> <span m="3193650">outcome.</span> <span m="3194910">This</span> <span m="3195070">is</span> <span m="3195170">y/n</span> <span m="3195880">times</span> <span m="3196180">1/2,</span> <span m="3196660">is</span> <span m="3196820">y</span> <span m="3197180">over</span> <span m="3197420">2n.</span> <span m="3199490">This</span> <span m="3200000">is</span> <span m="3200130">y</span> <span m="3200340">over</span> <span m="3200550">2n</span> <span m="3200930">also.</span> <span m="3202920">This</span> <span m="3203150">is</span> <span m="3203380">z</span> <span m="3203570">minus</span> <span m="3203950">y</span> <span m="3204390">over</span> <span m="3204630">2n.</span> <span m="3207742">z</span> <span m="3208220">minus</span> <span m="3208610">y</span> <span m="3208880">over</span> <span m="3209070">2n.</span> <span m="3211090">n</span> <span m="3211350">minus</span> <span m="3211710">z</span> <span m="3212200">over</span> <span m="3212410">2n.</span> <span m="3219298">All right, I</span> <span m="3219770">got</span> <span m="3219930">all</span> <span m="3220250">the</span> <span m="3221210">sample</span> <span m="3221590">points.</span> <span m="3222030">I</span> <span m="3222090">got</span> <span m="3222380">all</span> <span m="3222670">the</span> <span m="3223160">probabilities.</span> </p>
<p><span m="3225530">Let's</span> <span m="3225720">figure</span> <span m="3226020">out</span> <span m="3226230">if</span> <span m="3226330">you</span> <span m="3226470">win</span> <span m="3226730">now.</span> <span m="3227030">And</span> <span m="3227250">to</span> <span m="3227340">do</span> <span m="3227520">that,</span> <span m="3227890">the</span> <span m="3228260">first</span> <span m="3228520">thing</span> <span m="3228610">we have to</span> <span m="3228760">figure</span> <span m="3228980">out</span> <span m="3229080">is,</span> <span m="3229240">do</span> <span m="3229360">you</span> <span m="3229540">swap?</span> <span m="3232650">Do</span> <span m="3232790">you</span> <span m="3233000">swap</span> <span m="3233680">here?</span> <span m="3238670">Well,</span> <span m="3238890">what</span> <span m="3239020">happened?</span> <span m="3239460">I</span> <span m="3239530">revealed</span> <span m="3240020">y,</span> <span m="3241310">which</span> <span m="3241550">is</span> <span m="3241700">bigger</span> <span m="3243050">than</span> <span m="3243210">my</span> <span m="3243380">split</span> <span m="3243800">value</span> <span m="3244240">x,</span> <span m="3244530">my</span> <span m="3244660">guessed</span> <span m="3244950">value.</span> <span m="3246270">And</span> <span m="3246460">so</span> <span m="3246710">I</span> <span m="3246990">am</span> <span m="3247160">guessing</span> <span m="3247690">I've</span> <span m="3247820">got</span> <span m="3248380">the</span> <span m="3248490">biggest</span> <span m="3248830">value.</span> <span m="3250890">So</span> <span m="3251030">I</span> <span m="3251280">don't</span> <span m="3251650">swap.</span> <span m="3253800">So</span> <span m="3253960">there's</span> <span m="3254150">no</span> <span m="3254580">swap.</span> </p>
<p><span m="3258340">What</span> <span m="3258470">about</span> <span m="3258700">here?</span> <span m="3259060">Do</span> <span m="3259190">I</span> <span m="3259270">swap</span> <span m="3259690">here?</span> <span m="3263070">I</span> <span m="3263280">open</span> <span m="3263700">z,</span> <span m="3264260">I</span> <span m="3264360">saw</span> <span m="3264740">z.</span> <span m="3266060">z</span> <span m="3266500">is</span> <span m="3266640">bigger</span> <span m="3267160">than</span> <span m="3267360">what</span> <span m="3267490">I</span> <span m="3267560">think</span> <span m="3267770">the</span> <span m="3267860">midpoint</span> <span m="3268380">is,</span> <span m="3269890">so</span> <span m="3270030">I</span> <span m="3270100">wouldn't</span> <span m="3270420">swap.</span> <span m="3270740">I</span> <span m="3270930">only</span> <span m="3271300">guess</span> <span m="3272580">if</span> <span m="3272720">the</span> <span m="3272810">value</span> <span m="3273220">I</span> <span m="3273310">reveal</span> <span m="3273830">is</span> <span m="3274010">less</span> <span m="3274530">than</span> <span m="3274670">my</span> <span m="3274820">guessed</span> <span m="3275440">midpoint.</span> <span m="3277142">If</span> <span m="3277580">I</span> <span m="3277780">got</span> <span m="3278030">a value</span> <span m="3278420">I see</span> <span m="3278810">bigger</span> <span m="3279250">than</span> <span m="3279530">my</span> <span m="3279680">midpoint,</span> <span m="3280270">I</span> <span m="3280340">think</span> <span m="3280570">I've</span> <span m="3280670">got</span> <span m="3280870">the</span> <span m="3280950">big</span> <span m="3281150">one.</span> <span m="3282256">So</span> <span m="3282680">there's</span> <span m="3282910">no</span> <span m="3283230">swap.</span> <span m="3285870">What</span> <span m="3285990">about</span> <span m="3286200">here?</span> <span m="3286460">Do I</span> <span m="3286600">swap</span> <span m="3286940">here?</span> <span m="3288490">Yes,</span> <span m="3288930">here</span> <span m="3289150">I</span> <span m="3289220">swap</span> <span m="3289650">because</span> <span m="3289810">I</span> <span m="3289920">open</span> <span m="3290210">up</span> <span m="3290360">something</span> <span m="3290780">that's</span> <span m="3291000">less</span> <span m="3292270">than</span> <span m="3292380">my</span> <span m="3292520">guessed</span> <span m="3292830">midpoint,</span> <span m="3293120">so</span> <span m="3293410">I</span> <span m="3293460">think</span> <span m="3293650">I've</span> <span m="3293690">got</span> <span m="3293860">the</span> <span m="3293940">small</span> <span m="3294270">one</span> <span m="3294430">so</span> <span m="3294530">I'm</span> <span m="3294600">going</span> <span m="3294690">to</span> <span m="3294750">swap.</span> </p>
<p><span m="3299060">What</span> <span m="3299300">about</span> <span m="3299540">here?</span> <span m="3299750">Do</span> <span m="3299840">I</span> <span m="3299910">swap</span> <span m="3300220">here?</span> <span m="3301620">No</span> <span m="3301940">swap,</span> <span m="3302440">because</span> <span m="3302650">I</span> <span m="3302720">opened</span> <span m="3302980">up</span> <span m="3303080">something</span> <span m="3303340">that's</span> <span m="3303560">bigger</span> <span m="3303830">than</span> <span m="3303970">midpoint.</span> <span m="3305070">What</span> <span m="3305230">about</span> <span m="3305490">here?</span> <span m="3307080">I</span> <span m="3307220">swap.</span> <span m="3309760">And</span> <span m="3310000">what</span> <span m="3310130">about</span> <span m="3310340">here?</span> <span m="3311540">Swap</span> <span m="3312025">on</span> <span m="3312510">both of</span> <span m="3312580">them,</span> <span m="3312770">because</span> <span m="3313090">both</span> <span m="3313360">of them are</span> <span m="3313620">smaller</span> <span m="3314280">than</span> <span m="3314650">my</span> <span m="3314850">guessed</span> <span m="3315160">midpoint.</span> <span m="3317370">All</span> <span m="3317500">right,</span> <span m="3317680">now</span> <span m="3318190">let's</span> <span m="3318380">figure</span> <span m="3318600">out</span> <span m="3318700">if</span> <span m="3318790">you</span> <span m="3318920">won</span> <span m="3319170">or you</span> <span m="3319290">lost.</span> </p>
<p><span m="3321210">So</span> <span m="3321420">here</span> <span m="3321660">I</span> <span m="3321710">did</span> <span m="3321930">not</span> <span m="3322180">swap</span> <span m="3324650">and</span> <span m="3324830">I</span> <span m="3324920">started</span> <span m="3325350">with</span> <span m="3325510">a</span> <span m="3325570">smaller</span> <span m="3326010">value.</span> <span m="3326550">So</span> <span m="3326620">what</span> <span m="3326780">happened?</span> <span m="3327230">Did I</span> <span m="3327270">win</span> <span m="3327490">or</span> <span m="3327560">lose?</span> <span m="3329715">Lose.</span> <span m="3330850">I</span> <span m="3330980">opened he</span> <span m="3331300">smaller</span> <span m="3331700">value</span> <span m="3332010">and did</span> <span m="3332160">not</span> <span m="3332390">swap.</span> <span m="3333550">What</span> <span m="3333680">happened</span> <span m="3333960">here?</span> <span m="3335970">I</span> <span m="3336110">win.</span> <span m="3336650">I</span> <span m="3336790">open</span> <span m="3337070">the</span> <span m="3337160">bigger</span> <span m="3337560">value</span> <span m="3337880">and</span> <span m="3337950">did</span> <span m="3338040">not</span> <span m="3338300">swap.</span> <span m="3340020">Here</span> <span m="3340380">what</span> <span m="3340500">happened?</span> <span m="3342350">Win,</span> <span m="3342810">I</span> <span m="3342930">opened</span> <span m="3343180">the</span> <span m="3343250">small</span> <span m="3343640">value</span> <span m="3343940">when</span> <span m="3344050">I</span> <span m="3344110">swapped,</span> <span m="3344640">that's</span> <span m="3344870">a</span> <span m="3344940">win.</span> <span m="3345950">Here?</span> <span m="3348330">I</span> <span m="3348460">opened</span> <span m="3348750">the</span> <span m="3348820">big</span> <span m="3349180">value,</span> <span m="3349870">did</span> <span m="3350080">not</span> <span m="3350370">swap.</span> <span m="3351280">It's</span> <span m="3351420">a</span> <span m="3351490">win.</span> <span m="3353590">Here</span> <span m="3353870">I</span> <span m="3353950">opened</span> <span m="3354210">a</span> <span m="3354270">small</span> <span m="3354730">value</span> <span m="3355150">and</span> <span m="3355230">swap,</span> <span m="3355780">that's</span> <span m="3355980">a</span> <span m="3356050">win.</span> <span m="3357590">And</span> <span m="3357770">here</span> <span m="3358090">I</span> <span m="3358170">opened</span> <span m="3358560">a</span> <span m="3358790">big</span> <span m="3359220">value</span> <span m="3359770">and</span> <span m="3359900">swap,</span> <span m="3360320">that's</span> <span m="3360540">a</span> <span m="3360600">lose.</span> </p>
<p><span m="3362812">All</span> <span m="3363220">right,</span> <span m="3363420">now</span> <span m="3363670">we</span> <span m="3363780">can</span> <span m="3363900">compute</span> <span m="3364460">the</span> <span m="3364560">probability</span> <span m="3365120">of</span> <span m="3365190">winning.</span> <span m="3368240">The</span> <span m="3368320">probability</span> <span m="3368840">of</span> <span m="3368910">a</span> <span m="3368980">win</span> <span m="3370820">is</span> <span m="3371030">the</span> <span m="3371140">sum</span> <span m="3371560">of</span> <span m="3371670">these</span> <span m="3371950">four</span> <span m="3372160">sample</span> <span m="3372540">points--</span> <span m="3374220">y</span> <span m="3374580">over</span> <span m="3374920">2n</span> <span m="3376320">plus</span> <span m="3376640">z</span> <span m="3376860">minus</span> <span m="3377310">y</span> <span m="3378440">over</span> <span m="3378630">2n</span> <span m="3379950">plus</span> <span m="3382110">z</span> <span m="3382310">minus</span> <span m="3382710">y--</span> <span m="3383540">whoops--</span> <span m="3385610">over</span> <span m="3385740">2n</span> <span m="3386620">plus</span> <span m="3388050">n</span> <span m="3388300">minus</span> <span m="3388640">z</span> <span m="3388840">over</span> <span m="3389010">2n.</span> <span m="3393090">That</span> <span m="3393310">equals--</span> <span m="3395390">z</span> <span m="3395760">cancels</span> <span m="3396140">the</span> <span m="3396240">z, and</span> <span m="3396665">a y</span> <span m="3397090">cancels</span> <span m="3397510">the</span> <span m="3397590">y,</span> <span m="3398830">so</span> <span m="3399010">I've</span> <span m="3399210">got</span> <span m="3401400">n,</span> <span m="3402170">one</span> <span m="3402370">left,</span> <span m="3402680">n</span> <span m="3402860">left,</span> <span m="3403170">one</span> <span m="3403500">z</span> <span m="3403830">left,</span> <span m="3404820">and</span> <span m="3404970">one</span> <span m="3405210">negative</span> <span m="3405550">y</span> <span m="3405830">left</span> <span m="3406770">over</span> <span m="3407000">2n.</span> <span m="3409730">And</span> <span m="3409820">I</span> <span m="3409870">can</span> <span m="3410050">simplify</span> <span m="3410640">that.</span> <span m="3410990">N</span> <span m="3411160">over</span> <span m="3411350">2n</span> <span m="3411700">is</span> <span m="3411970">1/2,</span> <span m="3414080">plus</span> <span m="3414650">now</span> <span m="3414840">what's</span> <span m="3415020">left</span> <span m="3415260">over is</span> <span m="3415580">z</span> <span m="3415930">minus</span> <span m="3416370">y</span> <span m="3417680">over</span> <span m="3419180">2n.</span> <span m="3422140">And</span> <span m="3422290">we</span> <span m="3422380">know</span> <span m="3422580">the</span> <span m="3422740">z</span> <span m="3422940">and</span> <span m="3423050">y are</span> <span m="3423440">different</span> <span m="3423840">by</span> <span m="3424000">at</span> <span m="3424160">least</span> <span m="3424650">1.</span> </p>
<p><span m="3426960">So</span> <span m="3427120">this</span> <span m="3427330">is</span> <span m="3427450">at</span> <span m="3427580">least</span> <span m="3428790">1/2</span> <span m="3430080">plus</span> <span m="3430530">1</span> <span m="3430790">over</span> <span m="3431010">2n.</span> <span m="3434360">And</span> <span m="3434510">so</span> <span m="3434620">if n is</span> <span m="3435000">100,</span> <span m="3436390">you've</span> <span m="3436500">got</span> <span m="3436680">a</span> <span m="3436740">50</span> <span m="3437110">and</span> <span m="3437490">1/2%</span> <span m="3438950">chance</span> <span m="3439190">of</span> <span m="3439270">winning.</span> <span m="3440280">If</span> <span m="3440430">n</span> <span m="3440580">is</span> <span m="3440690">10,</span> <span m="3442300">the</span> <span m="3442390">numbers</span> <span m="3442710">are</span> <span m="3442770">from</span> <span m="3442920">0</span> <span m="3443210">to</span> <span m="3443310">10,</span> <span m="3444930">you've</span> <span m="3445030">got</span> <span m="3445170">at</span> <span m="3445330">least</span> <span m="3445630">a</span> <span m="3445680">55%</span> <span m="3446780">chance</span> <span m="3447080">to</span> <span m="3447190">win,</span> <span m="3448220">which</span> <span m="3448390">is</span> <span m="3448500">pretty</span> <span m="3448690">high,</span> <span m="3451030">OK?</span> <span m="3453990">Any</span> <span m="3454550">questions</span> <span m="3455040">here?</span> <span m="3455750">You</span> <span m="3455830">see</span> <span m="3456030">what's</span> <span m="3456220">going</span> <span m="3456420">on?</span> <span m="3456700">So</span> <span m="3456750">here's</span> <span m="3457050">the</span> <span m="3457140">zone</span> <span m="3457430">where</span> <span m="3457520">you</span> <span m="3457620">guessed</span> <span m="3457930">right.</span> <span m="3459660">You</span> <span m="3459830">get</span> <span m="3459990">the</span> <span m="3460090">win</span> <span m="3460320">either</span> <span m="3460540">way.</span> <span m="3461190">Here</span> <span m="3461460">you</span> <span m="3461730">guess</span> <span m="3462020">low,</span> <span m="3462490">and it</span> <span m="3462670">doesn't</span> <span m="3462910">make</span> <span m="3463100">any</span> <span m="3463240">difference.</span> <span m="3463630">You're</span> <span m="3463740">50-50.</span> <span m="3464940">You</span> <span m="3465050">guessed</span> <span m="3465370">high</span> <span m="3466240">and</span> <span m="3466330">you're</span> <span m="3466430">50-50</span> <span m="3466825">again.</span> <span m="3468690">So</span> <span m="3468880">guessing</span> <span m="3470260">helps.</span> <span m="3472920">Any</span> <span m="3473060">questions</span> <span m="3473710">about</span> <span m="3473960">that?</span> <span m="3474500">Yeah.</span> </p>
<p><span m="3475532">AUDIENCE: So aren't</span> <span m="3476024">you assuming</span> <span m="3476516">that--</span> <span m="3477008">because you</span> <span m="3477500">say,</span> <span m="3478484">have</span> <span m="3478976">a range,</span> <span m="3479960">a greater</span> <span m="3480452">range of</span> <span m="3480944">numbers,</span> <span m="3481436">that's there's</span> <span m="3481928">a greater</span> <span m="3482420">chance</span> <span m="3482912">that</span> <span m="3483404">the</span> <span m="3483896">number is going</span> <span m="3484388">fall in that</span> <span m="3484880">range.</span> </p>
<p><span m="3487840">PROFESSOR: I</span> <span m="3487970">was</span> <span m="3488190">the</span> <span m="3488270">nasty</span> <span m="3488610">guy</span> <span m="3488910">here.</span> <span m="3490050">I</span> <span m="3490210">picked</span> <span m="3490440">z</span> <span m="3490710">and</span> <span m="3490800">y</span> <span m="3491180">to</span> <span m="3491270">be</span> <span m="3491410">consecutive</span> <span m="3491990">numbers,</span> <span m="3492500">which</span> <span m="3492720">minimized</span> <span m="3493960">your</span> <span m="3494130">chance</span> <span m="3495520">of</span> <span m="3495690">doing</span> <span m="3495920">well</span> <span m="3496220">with this</span> <span m="3496340">strategy.</span> </p>
<p><span m="3497000">AUDIENCE: Because it</span> <span m="3497483">seems like</span> <span m="3497966">we don't know</span> <span m="3498449">anything about</span> <span m="3498932">[INAUDIBLE].</span> </p>
<p><span m="3502796">PROFESSOR: The distribution of</span> <span m="3503279">what?</span> </p>
<p><span m="3504245">AUDIENCE: Of</span> <span m="3505710">possible</span> <span m="3506150">other numbers</span> <span m="3506590">in</span> <span m="3506933">envelopes</span> <span m="3507276">[INAUDIBLE].</span> </p>
<p><span m="3507620">PROFESSOR: Right,</span> <span m="3508290">you</span> <span m="3508500">know</span> <span m="3508730">nothing</span> <span m="3509150">about</span> <span m="3509340">the</span> <span m="3509420">distribution</span> <span m="3510000">because</span> <span m="3510290">there</span> <span m="3510450">is</span> <span m="3510670">none.</span> <span m="3512120">There's</span> <span m="3512360">no</span> <span m="3512690">randomness</span> <span m="3513260">there.</span> <span m="3513530">I</span> <span m="3513770">picked</span> <span m="3514220">worst</span> <span m="3514560">case</span> <span m="3514850">numbers</span> <span m="3515230">here.</span> </p>
<p><span m="3516900">AUDIENCE: But it's</span> <span m="3517860">still</span> <span m="3518340">sort of like</span> <span m="3518820">a</span> <span m="3519300">[INAUDIBLE].</span> <span m="3521700">If</span> <span m="3522180">it's 10,</span> <span m="3522660">you still have</span> <span m="3523140">a better chance</span> <span m="3523620">to</span> <span m="3524100">swap</span> <span m="3525060">than</span> <span m="3525540">if you</span> <span m="3526020">[INAUDIBLE]</span> <span m="3526980">below</span> <span m="3527480">that.</span> </p>
<p><span m="3528120">PROFESSOR: No.</span> <span m="3529460">No,</span> <span m="3530060">any</span> <span m="3530360">deterministic</span> <span m="3531120">strategy</span> <span m="3531570">you</span> <span m="3531700">pick,</span> <span m="3532320">if</span> <span m="3532440">you</span> <span m="3532530">don't</span> <span m="3532720">guess</span> <span m="3532950">that</span> <span m="3533130">random</span> <span m="3533460">x,</span> <span m="3534470">you</span> <span m="3534640">will</span> <span m="3534780">not</span> <span m="3535060">do</span> <span m="3535170">better</span> <span m="3535370">than</span> <span m="3535510">50-50.</span> <span m="3538110">The</span> <span m="3538240">only</span> <span m="3538600">way</span> <span m="3538800">you'd</span> <span m="3538960">beat</span> <span m="3539200">50-50</span> <span m="3539940">is</span> <span m="3540110">to,</span> <span m="3540580">in</span> <span m="3540730">your</span> <span m="3540860">mind,</span> <span m="3541470">make</span> <span m="3541540">a</span> <span m="3541690">random</span> <span m="3542150">number</span> <span m="3543600">13</span> <span m="3544000">and</span> <span m="3544400">1/2</span> <span m="3545580">and</span> <span m="3545810">flip</span> <span m="3546190">if</span> <span m="3546690">you</span> <span m="3547390">see</span> <span m="3547680">less</span> <span m="3547890">than</span> <span m="3548010">13</span> <span m="3548420">and</span> <span m="3548830">1/2.</span> <span m="3549780">If</span> <span m="3549910">you</span> <span m="3550060">come</span> <span m="3550260">in</span> <span m="3550370">with</span> <span m="3550510">a</span> <span m="3550560">strategy</span> <span m="3551090">that</span> <span m="3551250">hey,</span> <span m="3551640">10</span> <span m="3552030">is</span> <span m="3552150">a</span> <span m="3552220">small</span> <span m="3552660">number,</span> <span m="3554580">out</span> <span m="3554750">of</span> <span m="3554820">100,</span> <span m="3555350">and</span> <span m="3555440">so</span> <span m="3555540">I'm</span> <span m="3555680">going</span> <span m="3555780">to</span> <span m="3555840">swap</span> <span m="3556470">if</span> <span m="3556610">I</span> <span m="3556690">see</span> <span m="3556920">a</span> <span m="3556990">10,</span> <span m="3559380">well,</span> <span m="3559640">no,</span> <span m="3559980">because</span> <span m="3560740">first</span> <span m="3561040">it</span> <span m="3561100">could</span> <span m="3561220">be--</span> <span m="3561690">I</span> <span m="3561810">might</span> <span m="3562100">have</span> <span m="3562370">done</span> <span m="3562600">9</span> <span m="3562870">or</span> <span m="3562950">11,</span> <span m="3563350">either</span> <span m="3563530">one.</span> <span m="3563720">In fact,</span> <span m="3564080">if I</span> <span m="3564230">know</span> <span m="3564530">that's</span> <span m="3564710">your</span> <span m="3564870">strategy,</span> <span m="3566470">I'm</span> <span m="3566640">going</span> <span m="3566760">to</span> <span m="3566810">make</span> <span m="3567000">a</span> <span m="3567060">9.</span> <span m="3569530">And</span> <span m="3569710">then</span> <span m="3569840">you're</span> <span m="3569960">doomed</span> <span m="3570360">in</span> <span m="3570490">seeing</span> <span m="3570880">a</span> <span m="3570940">10.</span> </p>
<p><span m="3572620">AUDIENCE: So is there</span> <span m="3573080">a best way</span> <span m="3573540">to pick that?</span> </p>
<p><span m="3574870">PROFESSOR: Yes,</span> <span m="3575630">I also</span> <span m="3576005">have</span> <span m="3576380">an optimal</span> <span m="3576870">strategy.</span> <span m="3578680">In</span> <span m="3578800">fact,</span> <span m="3580280">there's</span> <span m="3580480">two</span> <span m="3580830">interesting</span> <span m="3581140">things</span> <span m="3581350">about</span> <span m="3581570">this</span> <span m="3581760">game</span> <span m="3581960">which</span> <span m="3582090">we</span> <span m="3582200">won't</span> <span m="3582440">prove.</span> <span m="3583630">The</span> <span m="3583720">first</span> <span m="3584000">thing</span> <span m="3584110">we'll</span> <span m="3584230">prove,</span> <span m="3584900">we</span> <span m="3585160">will</span> <span m="3585280">prove</span> <span m="3585610">but</span> <span m="3585740">it's</span> <span m="3585830">true,</span> <span m="3586500">is</span> <span m="3586770">this</span> <span m="3587150">is</span> <span m="3587340">the</span> <span m="3587670">optimal</span> <span m="3588030">strategy</span> <span m="3588570">for</span> <span m="3588710">you.</span> <span m="3590470">And</span> <span m="3590630">my</span> <span m="3590960">optimal</span> <span m="3591370">strategy</span> <span m="3592510">is</span> <span m="3593050">to</span> <span m="3593170">pick</span> <span m="3593520">a</span> <span m="3593640">random</span> <span m="3594520">value</span> <span m="3594820">of</span> <span m="3594950">y</span> <span m="3595370">between</span> <span m="3595730">0</span> <span m="3596080">and</span> <span m="3596520">n</span> <span m="3596650">minus</span> <span m="3596950">1,</span> <span m="3597820">and</span> <span m="3597930">then</span> <span m="3598080">to</span> <span m="3598140">make</span> <span m="3598360">z</span> <span m="3598660">be</span> <span m="3598790">one</span> <span m="3599020">more.</span> <span m="3601560">And</span> <span m="3601660">the</span> <span m="3601740">way</span> <span m="3601880">to</span> <span m="3601980">think</span> <span m="3602220">about</span> <span m="3602500">this,</span> <span m="3602750">and</span> <span m="3602850">there's</span> <span m="3603050">whole</span> <span m="3603340">classes</span> <span m="3603790">you</span> <span m="3603950">can</span> <span m="3604110">take</span> <span m="3604360">on</span> <span m="3604520">game</span> <span m="3604850">theory,</span> <span m="3606520">is</span> <span m="3606780">that</span> <span m="3606900">suppose</span> <span m="3607520">I</span> <span m="3607720">do</span> <span m="3608040">pick</span> <span m="3609300">my</span> <span m="3609560">guys</span> <span m="3610660">randomly</span> <span m="3611170">by</span> <span m="3611280">picking</span> <span m="3611600">a</span> <span m="3611680">random</span> <span m="3612080">y</span> <span m="3612320">uniformly</span> <span m="3613610">from</span> <span m="3613750">0</span> <span m="3614000">to</span> <span m="3614080">n</span> <span m="3614250">minus</span> <span m="3614560">1.</span> <span m="3614900">So</span> <span m="3615000">my</span> <span m="3615180">smaller</span> <span m="3615570">number</span> <span m="3615960">is</span> <span m="3616750">anything</span> <span m="3617080">between</span> <span m="3617340">0</span> <span m="3617610">and</span> <span m="3617680">99,</span> <span m="3618510">equally</span> <span m="3618830">likely,</span> <span m="3619960">and then</span> <span m="3620180">the</span> <span m="3620250">bigger</span> <span m="3620520">number's</span> <span m="3620850">just</span> <span m="3621060">one</span> <span m="3621560">more.</span> </p>
<p><span m="3622550">So</span> <span m="3622720">if I</span> <span m="3622810">picked</span> <span m="3623120">92</span> <span m="3623660">for</span> <span m="3623750">the</span> <span m="3623860">random</span> <span m="3624180">one,</span> <span m="3624340">the</span> <span m="3624420">next</span> <span m="3624650">one's</span> <span m="3624750">going to</span> <span m="3624890">be</span> <span m="3624970">93.</span> <span m="3626360">If</span> <span m="3626520">that's</span> <span m="3626770">my</span> <span m="3626980">strategy,</span> <span m="3628320">even</span> <span m="3628600">if</span> <span m="3628790">I</span> <span m="3628900">tell</span> <span m="3629190">you</span> <span m="3629340">that,</span> <span m="3630350">you</span> <span m="3630600">cannot</span> <span m="3631070">do</span> <span m="3631280">any</span> <span m="3631510">better</span> <span m="3632960">than</span> <span m="3633130">getting</span> <span m="3633440">1/2</span> <span m="3634000">plus</span> <span m="3634180">1</span> <span m="3634360">over</span> <span m="3634530">2n</span> <span m="3635510">probability</span> <span m="3635970">of</span> <span m="3636040">winning,</span> <span m="3636340">no</span> <span m="3636420">matter</span> <span m="3636660">what</span> <span m="3637080">you</span> <span m="3637280">do</span> <span m="3637460">in</span> <span m="3637530">the</span> <span m="3637610">whole</span> <span m="3637770">world,</span> <span m="3639470">OK?</span> <span m="3640420">And</span> <span m="3640610">if</span> <span m="3640710">you</span> <span m="3640880">use</span> <span m="3641210">this</span> <span m="3641430">strategy</span> <span m="3642040">here,</span> <span m="3643620">no</span> <span m="3643720">matter</span> <span m="3643990">what</span> <span m="3644490">I</span> <span m="3644660">do,</span> <span m="3645740">even</span> <span m="3646070">knowing</span> <span m="3646380">that's</span> <span m="3646540">your</span> <span m="3646690">strategy,</span> <span m="3647950">I</span> <span m="3648090">can't</span> <span m="3648500">keep</span> <span m="3648840">you</span> <span m="3649680">from</span> <span m="3649870">getting</span> <span m="3650930">this</span> <span m="3651140">much.</span> <span m="3652730">It's</span> <span m="3652820">called</span> <span m="3653030">a</span> <span m="3653060">minimax</span> <span m="3653690">solution.</span> </p>
<p><span m="3655620">So</span> <span m="3656150">my</span> <span m="3656390">optimal</span> <span m="3656710">strategy</span> <span m="3657210">is</span> <span m="3657290">to</span> <span m="3657370">pick</span> <span m="3657560">uniformly</span> <span m="3658170">in</span> <span m="3658260">0</span> <span m="3658520">to</span> <span m="3658600">99,</span> <span m="3659170">and</span> <span m="3659250">the</span> <span m="3659330">next</span> <span m="3659590">one's</span> <span m="3659720">one</span> <span m="3659900">more.</span> <span m="3660470">Your</span> <span m="3660740">optimal</span> <span m="3661050">strategy</span> <span m="3661600">is</span> <span m="3661890">this.</span> <span m="3662880">Pick</span> <span m="3663150">a</span> <span m="3664260">number</span> <span m="3664590">uniformly</span> <span m="3664960">in the</span> <span m="3665330">half</span> <span m="3665600">integers</span> <span m="3665990">there,</span> <span m="3666550">and</span> <span m="3666690">swap</span> <span m="3667050">if</span> <span m="3667140">you</span> <span m="3667240">see</span> <span m="3667380">something</span> <span m="3667640">smaller.</span> <span m="3668030">And</span> <span m="3668110">there's</span> <span m="3668290">no</span> <span m="3668550">better</span> <span m="3668800">strategy.</span> <span m="3670310">In</span> <span m="3670410">fact,</span> <span m="3670670">any</span> <span m="3670840">deterministic</span> <span m="3671430">strategy,</span> <span m="3672270">you</span> <span m="3672370">don't</span> <span m="3672580">do</span> <span m="3672670">better</span> <span m="3672830">than</span> <span m="3672950">50-50.</span> <span m="3674220">The</span> <span m="3674370">optimal</span> <span m="3674760">strategies</span> <span m="3675310">require</span> <span m="3676030">randomness</span> <span m="3677840">for</span> <span m="3678020">each</span> <span m="3678160">of</span> <span m="3678240">us.</span> <span m="3680640">OK,</span> <span m="3680910">any</span> <span m="3682030">more</span> <span m="3682250">questions</span> <span m="3682700">about</span> <span m="3684250">what</span> <span m="3684750">happened</span> <span m="3685120">here,</span> <span m="3686410">this</span> <span m="3686600">game?</span> <span m="3687320">Yeah.</span> </p>
<p><span m="3688642">AUDIENCE: [INAUDIBLE]</span> <span m="3689604">when you</span> <span m="3690085">see a small number</span> <span m="3690566">you're going</span> <span m="3691047">to swap,</span> <span m="3691528">does</span> <span m="3692009">that</span> <span m="3692490">just mean</span> <span m="3692971">[INAUDIBLE]?</span> </p>
<p><span m="3695380">PROFESSOR: Yeah,</span> <span m="3696160">basically,</span> <span m="3697510">yeah,</span> <span m="3697870">that's</span> <span m="3698040">what</span> <span m="3698150">it</span> <span m="3698200">means.</span> <span m="3699220">Effectively,</span> <span m="3699750">that</span> <span m="3699970">50</span> <span m="3700420">is</span> <span m="3700550">your</span> <span m="3700880">random</span> <span m="3701240">number.</span> <span m="3701650">If</span> <span m="3701780">you</span> <span m="3701900">think</span> <span m="3702160">OK,</span> <span m="3702360">if</span> <span m="3702460">I</span> <span m="3702520">see</span> <span m="3702710">less</span> <span m="3702940">than</span> <span m="3703060">50,</span> <span m="3703360">I'm</span> <span m="3703420">going</span> <span m="3703500">to</span> <span m="3703550">swap,</span> <span m="3704180">bigger</span> <span m="3704400">than</span> <span m="3704540">50</span> <span m="3704820">I</span> <span m="3704880">don't.</span> <span m="3705240">Now you</span> <span m="3705700">have</span> <span m="3705820">to</span> <span m="3705920">decide</span> <span m="3706250">what</span> <span m="3706370">happens</span> <span m="3706710">at</span> <span m="3706850">50,</span> <span m="3707780">if</span> <span m="3707890">you</span> <span m="3707990">saw</span> <span m="3708240">exactly</span> <span m="3708690">50.</span> <span m="3709010">So</span> <span m="3709150">you'd</span> <span m="3709260">probably</span> <span m="3709640">go</span> <span m="3709840">in</span> <span m="3710340">at</span> <span m="3710490">49</span> <span m="3710990">and</span> <span m="3711050">1/2</span> <span m="3711370">and</span> <span m="3711490">50</span> <span m="3711860">and 1/2.</span> <span m="3712860">And</span> <span m="3713030">so</span> <span m="3713130">that</span> <span m="3713370">could</span> <span m="3713600">be</span> <span m="3713750">construed</span> <span m="3714130">as</span> <span m="3714220">that</span> <span m="3714420">way.</span> <span m="3714610">But</span> <span m="3714720">you</span> <span m="3714810">didn't</span> <span m="3715040">pick</span> <span m="3715220">it</span> <span m="3715330">randomly,</span> <span m="3716790">and</span> <span m="3717060">that</span> <span m="3717220">sort</span> <span m="3717400">of</span> <span m="3717470">human</span> <span m="3717750">intuition,</span> <span m="3719280">to</span> <span m="3719410">swap</span> <span m="3719680">so</span> <span m="3719810">I</span> <span m="3719950">could</span> <span m="3720130">use</span> <span m="3720370">that</span> <span m="3720550">now</span> <span m="3720680">to</span> <span m="3720750">design</span> <span m="3721070">my</span> <span m="3721240">strategy,</span> <span m="3722980">knowing</span> <span m="3723440">that</span> <span m="3724260">your</span> <span m="3724720">random</span> <span m="3725020">numbers</span> <span m="3725330">could</span> <span m="3725410">be</span> <span m="3725550">50</span> <span m="3725950">and 1/2,</span> <span m="3727330">OK?</span> </p>
<p><span m="3730500">No,</span> <span m="3730640">I'll</span> <span m="3730750">be</span> <span m="3730870">50-50</span> <span m="3731440">because</span> <span m="3731670">what</span> <span m="3731820">I'll</span> <span m="3732010">do,</span> <span m="3732160">if</span> <span m="3732270">I</span> <span m="3732350">know</span> <span m="3732490">that's</span> <span m="3732630">your</span> <span m="3732760">strategy,</span> <span m="3733740">my</span> <span m="3733910">numbers</span> <span m="3734240">will</span> <span m="3734320">be</span> <span m="3734430">1</span> <span m="3734660">and</span> <span m="3734770">2.</span> <span m="3737760">Well, when</span> <span m="3738230">you</span> <span m="3738400">do</span> <span m="3738820">these</span> <span m="3739000">analyses,</span> <span m="3739480">you</span> <span m="3739590">sort</span> <span m="3739850">of,</span> <span m="3740300">if</span> <span m="3740820">it's</span> <span m="3740970">declared</span> <span m="3741480">that's</span> <span m="3741690">it,</span> <span m="3742360">I</span> <span m="3742450">can</span> <span m="3742600">assume</span> <span m="3742980">that.</span> <span m="3743540">And</span> <span m="3743770">I</span> <span m="3743800">might</span> <span m="3744020">actually</span> <span m="3744360">guess</span> <span m="3744800">that.</span> <span m="3745486">In fact,</span> <span m="3745830">that's</span> <span m="3746000">why</span> <span m="3746160">I</span> <span m="3746260">did</span> <span m="3746740">pick</span> <span m="3747030">two</span> <span m="3747220">very</span> <span m="3747480">small</span> <span m="3747730">numbers</span> <span m="3747980">and</span> <span m="3748030">two</span> <span m="3748150">very</span> <span m="3748390">big</span> <span m="3748600">numbers.</span> <span m="3749590">But</span> <span m="3749820">given</span> <span m="3750140">the</span> <span m="3750220">perverse</span> <span m="3750610">nature</span> <span m="3750980">here,</span> <span m="3751170">when</span> <span m="3751270">you</span> <span m="3751340">saw</span> <span m="3751540">the</span> <span m="3751640">big</span> <span m="3751890">number,</span> <span m="3752560">you</span> <span m="3752820">got</span> <span m="3753000">rid</span> <span m="3753160">of</span> <span m="3753260">it,</span> <span m="3753830">and</span> <span m="3753960">when</span> <span m="3754030">you</span> <span m="3754110">saw</span> <span m="3754270">the</span> <span m="3754390">small</span> <span m="3754700">number,</span> <span m="3756620">you</span> <span m="3756770">kept</span> <span m="3757070">it.</span> <span m="3757780">So</span> <span m="3758030">you</span> <span m="3758240">sort of</span> <span m="3758410">did</span> <span m="3758620">reverse</span> <span m="3759050">psychology</span> <span m="3759820">and</span> <span m="3760610">it</span> <span m="3760750">worked</span> <span m="3761040">out,</span> <span m="3761280">by</span> <span m="3761410">chance,</span> <span m="3761950">that</span> <span m="3762070">it</span> <span m="3762120">worked</span> <span m="3762380">out</span> <span m="3762530">that.</span> <span m="3763900">Normally</span> <span m="3764170">it</span> <span m="3764440">would</span> <span m="3764560">go</span> <span m="3764710">the</span> <span m="3764820">other</span> <span m="3764970">way,</span> <span m="3766106">I</span> <span m="3766520">think.</span> <span m="3767272">Of</span> <span m="3767610">course,</span> <span m="3767760">you've</span> <span m="3767890">all</span> <span m="3768070">seen</span> <span m="3768230">enough</span> <span m="3768430">games</span> <span m="3768760">now,</span> <span m="3768880">you</span> <span m="3769140">know to</span> <span m="3769400">do</span> <span m="3769560">the</span> <span m="3769700">non-standard</span> <span m="3770980">thing,</span> <span m="3771220">I</span> <span m="3771270">think.</span> </p>
<p><span m="3773800">Now,</span> <span m="3774580">this</span> <span m="3774790">kind</span> <span m="3774960">of</span> <span m="3775110">thinking</span> <span m="3775470">comes</span> <span m="3775710">up</span> <span m="3775920">all</span> <span m="3776150">the</span> <span m="3776250">time</span> <span m="3776570">in</span> <span m="3776640">computer</span> <span m="3776910">science</span> <span m="3777230">algorithms.</span> <span m="3777700">You</span> <span m="3777790">know,</span> <span m="3778340">the</span> <span m="3778450">protocol</span> <span m="3779080">that's</span> <span m="3779260">used</span> <span m="3779550">to</span> <span m="3779640">communicate</span> <span m="3780170">across</span> <span m="3781270">a</span> <span m="3781350">network,</span> <span m="3781860">ethernet,</span> <span m="3782720">shared</span> <span m="3783130">bus</span> <span m="3783890">is</span> <span m="3784050">randomized.</span> <span m="3785750">Each</span> <span m="3786020">entity</span> <span m="3786370">that</span> <span m="3786490">wants</span> <span m="3786740">to</span> <span m="3786810">use</span> <span m="3787130">the</span> <span m="3787220">bus</span> <span m="3787710">flips</span> <span m="3788020">a</span> <span m="3788090">random</span> <span m="3788430">number,</span> <span m="3789060">flips</span> <span m="3789160">a</span> <span m="3789310">random</span> <span m="3789590">coin.</span> <span m="3790040">You</span> <span m="3790110">could</span> <span m="3790230">think</span> <span m="3790400">of</span> <span m="3790500">it</span> <span m="3790600">that</span> <span m="3790700">way.</span> <span m="3791430">And</span> <span m="3791570">it</span> <span m="3791600">broadcasts</span> <span m="3792340">with</span> <span m="3792760">that</span> <span m="3792920">probability.</span> <span m="3794270">And</span> <span m="3794660">if</span> <span m="3794730">there's</span> <span m="3794890">a</span> <span m="3794950">collision,</span> <span m="3795370">it</span> <span m="3795470">backs</span> <span m="3795810">off</span> <span m="3796150">and chooses</span> <span m="3796490">a</span> <span m="3796550">smaller</span> <span m="3796950">probability</span> <span m="3797440">next</span> <span m="3797680">time.</span> <span m="3798530">And</span> <span m="3798750">if</span> <span m="3798830">it</span> <span m="3798930">gets</span> <span m="3799150">through,</span> <span m="3799960">then</span> <span m="3800110">it</span> <span m="3800180">tries</span> <span m="3800450">to</span> <span m="3800560">cram</span> <span m="3800750">a</span> <span m="3800800">bunch</span> <span m="3801050">of</span> <span m="3801130">stuff</span> <span m="3801400">through,</span> <span m="3801800">until</span> <span m="3802040">you</span> <span m="3802120">get</span> <span m="3802240">a</span> <span m="3802290">collision</span> <span m="3802700">again.</span> </p>
<p><span m="3804060">The</span> <span m="3804240">best</span> <span m="3804560">algorithms</span> <span m="3805070">for</span> <span m="3805200">sorting</span> <span m="3805830">a</span> <span m="3805910">list</span> <span m="3806130">of</span> <span m="3806210">numbers</span> <span m="3806680">are</span> <span m="3806860">randomized,</span> <span m="3807460">quick</span> <span m="3807710">sort,</span> <span m="3807970">which</span> <span m="3808140">you'll</span> <span m="3808250">see</span> <span m="3808420">in</span> <span m="3808540">6046,</span> <span m="3809780">is</span> <span m="3809950">a</span> <span m="3810030">randomized</span> <span m="3810970">algorithm.</span> <span m="3811800">In</span> <span m="3811910">fact,</span> <span m="3812300">it's</span> <span m="3812470">not</span> <span m="3812750">dissimilar.</span> <span m="3813170">You</span> <span m="3813310">guess</span> <span m="3814300">a</span> <span m="3814460">random</span> <span m="3814850">value</span> <span m="3815290">and</span> <span m="3815410">split</span> <span m="3815800">all</span> <span m="3816000">the</span> <span m="3816090">numbers</span> <span m="3817100">based</span> <span m="3817360">on</span> <span m="3817440">the</span> <span m="3817530">random</span> <span m="3817800">value.</span> <span m="3818100">Who's</span> <span m="3818360">bigger,</span> <span m="3818610">who's</span> <span m="3818760">smaller,</span> <span m="3819250">and then</span> <span m="3819400">you</span> <span m="3819510">[INAUDIBLE].</span> <span m="3821030">You</span> <span m="3821160">do</span> <span m="3822040">things</span> <span m="3822300">like</span> <span m="3822570">that.</span> </p>
<p><span m="3824670">OK,</span> <span m="3826780">that's all I'm</span> <span m="3827040">going</span> <span m="3827220">to</span> <span m="3827320">say</span> <span m="3827490">about</span> <span m="3827630">uniform</span> <span m="3828150">distributions.</span> <span m="3829900">Next</span> <span m="3830250">I</span> <span m="3830300">want</span> <span m="3830460">to</span> <span m="3830510">talk</span> <span m="3830820">about</span> <span m="3831760">the</span> <span m="3831890">binomial</span> <span m="3832560">distribution.</span> <span m="3834400">And</span> <span m="3834600">this</span> <span m="3834820">is</span> <span m="3836420">the</span> <span m="3836540">most</span> <span m="3836940">important</span> <span m="3838100">distribution</span> <span m="3839300">in</span> <span m="3839460">computer</span> <span m="3839880">science,</span> <span m="3840290">and</span> <span m="3840400">probably</span> <span m="3840900">the</span> <span m="3840990">most</span> <span m="3841300">important</span> <span m="3842510">in</span> <span m="3842690">all</span> <span m="3842900">of</span> <span m="3843000">discrete</span> <span m="3844060">calculations</span> <span m="3844520">in the</span> <span m="3844980">sciences.</span> <span m="3847730">In</span> <span m="3848090">continuous</span> <span m="3848560">distributions,</span> <span m="3849130">you</span> <span m="3849220">get</span> <span m="3849340">the</span> <span m="3849420">normal</span> <span m="3849790">distribution.</span> <span m="3850480">But,</span> <span m="3851230">for</span> <span m="3851440">discrete</span> <span m="3851880">problems,</span> <span m="3852760">the</span> <span m="3852830">binomial</span> <span m="3853400">is</span> <span m="3853490">the</span> <span m="3853560">most</span> <span m="3853800">important.</span> </p>
<p><span m="3856562">And</span> <span m="3857020">there's</span> <span m="3857210">two</span> <span m="3857400">versions.</span> <span m="3857910">There's</span> <span m="3858120">the</span> <span m="3858290">unbiased</span> <span m="3859520">binomial</span> <span m="3860000">distribution</span> <span m="3861820">and</span> <span m="3861930">then</span> <span m="3862060">there's</span> <span m="3862250">the</span> <span m="3862560">biased</span> <span m="3862810">one.</span> <span m="3868710">Now</span> <span m="3868900">this</span> <span m="3869060">one's</span> <span m="3869200">a</span> <span m="3869260">little</span> <span m="3869690">more</span> <span m="3869870">complicated.</span> <span m="3870170">You've</span> <span m="3870470">got</span> <span m="3870610">a</span> <span m="3870660">parameter</span> <span m="3871340">n</span> <span m="3871470">again,</span> <span m="3873100">and</span> <span m="3873310">the</span> <span m="3873390">distribution</span> <span m="3874050">function</span> <span m="3874460">on</span> <span m="3874620">k,</span> <span m="3875470">probability</span> <span m="3876180">of</span> <span m="3876250">getting</span> <span m="3876520">k</span> <span m="3876870">is</span> <span m="3877130">n</span> <span m="3877380">choose</span> <span m="3877780">k</span> <span m="3879350">times</span> <span m="3879670">2</span> <span m="3879690">to</span> <span m="3879830">the</span> <span m="3879980">minus</span> <span m="3880350">n.</span> <span m="3882000">And</span> <span m="3882190">n</span> <span m="3882390">is</span> <span m="3882810">at</span> <span m="3882880">least</span> <span m="3883180">1,</span> <span m="3884110">and</span> <span m="3884500">k</span> <span m="3884920">is</span> <span m="3885060">between</span> <span m="3885560">0</span> <span m="3886070">and</span> <span m="3886640">n.</span> </p>
<p><span m="3888560">And</span> <span m="3888690">then</span> <span m="3888840">for</span> <span m="3888940">the</span> <span m="3889060">general</span> <span m="3891250">binomial</span> <span m="3891750">distribution,</span> <span m="3901790">we</span> <span m="3901910">have</span> <span m="3902320">fn</span> <span m="3902560">of</span> <span m="3902820">k,</span> <span m="3903210">the</span> <span m="3903330">probability</span> <span m="3903810">of</span> <span m="3903870">getting</span> <span m="3904140">k</span> <span m="3905360">is</span> <span m="3905840">n</span> <span m="3906070">choose</span> <span m="3906430">k</span> <span m="3907710">times--</span> <span m="3908820">actually, it's</span> <span m="3909210">fnp,</span> <span m="3910270">there's another</span> <span m="3910660">parameter</span> <span m="3911080">p</span> <span m="3911350">here.</span> <span m="3912500">p</span> <span m="3912870">to</span> <span m="3912940">the</span> <span m="3913060">k</span> <span m="3913590">times</span> <span m="3913840">1</span> <span m="3914060">minus</span> <span m="3914410">p</span> <span m="3915590">to</span> <span m="3915670">the</span> <span m="3915870">n</span> <span m="3916070">minus</span> <span m="3916440">k</span> <span m="3916660">power.</span> <span m="3917585">And</span> <span m="3918050">p</span> <span m="3918300">is</span> <span m="3918450">a</span> <span m="3918500">value</span> <span m="3918800">between</span> <span m="3919150">0</span> <span m="3919460">to</span> <span m="3919520">1</span> <span m="3919850">typically.</span> <span m="3920640">It's a</span> <span m="3920700">probability</span> <span m="3921280">of</span> <span m="3921350">something</span> <span m="3921700">happening.</span> </p>
<p><span m="3923330">So</span> <span m="3923380">in</span> <span m="3923440">the</span> <span m="3923540">general</span> <span m="3923980">case,</span> <span m="3924320">you've</span> <span m="3924420">got</span> <span m="3924610">nnp.</span> <span m="3926560">The</span> <span m="3926730">unbiased</span> <span m="3927500">case</span> <span m="3927890">corresponds</span> <span m="3928680">to</span> <span m="3928780">when</span> <span m="3928950">p</span> <span m="3929270">is</span> <span m="3929390">1/2.</span> <span m="3930750">Because</span> <span m="3930980">if</span> <span m="3931090">p</span> <span m="3931350">is</span> <span m="3931470">1/2,</span> <span m="3931980">you</span> <span m="3932110">get</span> <span m="3932590">2</span> <span m="3932855">to</span> <span m="3933120">the</span> <span m="3933200">minus</span> <span m="3933560">k</span> <span m="3933940">and</span> <span m="3934270">2</span> <span m="3934330">the</span> <span m="3934760">minus</span> <span m="3935200">n</span> <span m="3935330">minus</span> <span m="3935650">k,</span> <span m="3935840">and</span> <span m="3935930">that's</span> <span m="3936150">just</span> <span m="3936690">2</span> <span m="3937120">to the</span> <span m="3937160">minus</span> <span m="3937510">n.</span> <span m="3938600">So</span> <span m="3938750">you</span> <span m="3938850">can</span> <span m="3939000">think</span> <span m="3939210">of</span> <span m="3939320">this</span> <span m="3939550">as</span> <span m="3939670">the</span> <span m="3939780">case</span> <span m="3940150">when</span> <span m="3940420">p is</span> <span m="3940600">1/2.</span> </p>
<p><span m="3946330">All</span> <span m="3946440">right,</span> <span m="3949020">to</span> <span m="3949100">give</span> <span m="3949240">you</span> <span m="3949300">an</span> <span m="3949360">example</span> <span m="3949830">why</span> <span m="3950100">this</span> <span m="3950300">is</span> <span m="3950390">so</span> <span m="3950560">important,</span> <span m="3951110">why it</span> <span m="3951310">comes</span> <span m="3951590">up</span> <span m="3951750">all</span> <span m="3951860">the</span> <span m="3951960">time.</span> <span m="3954230">Imagine</span> <span m="3954690">that</span> <span m="3954790">you've</span> <span m="3954930">got</span> <span m="3955170">a</span> <span m="3955230">system</span> <span m="3955820">with</span> <span m="3956220">n</span> <span m="3956540">components.</span> <span m="3959030">And</span> <span m="3959450">that</span> <span m="3959640">each</span> <span m="3959950">component</span> <span m="3960540">fails</span> <span m="3961090">with</span> <span m="3961190">probability</span> <span m="3961800">p.</span> <span m="3963159">And you</span> <span m="3963612">want to</span> <span m="3964065">know the</span> <span m="3964520">probability</span> <span m="3965210">of</span> <span m="3965270">having</span> <span m="3965590">some</span> <span m="3966020">number</span> <span m="3967040">of</span> <span m="3967230">failures.</span> <span m="3975160">So</span> <span m="3975300">for</span> <span m="3975400">example,</span> <span m="3975900">there's</span> <span m="3976250">n</span> <span m="3976470">components</span> <span m="3982890">and</span> <span m="3983650">each</span> <span m="3987790">fails</span> <span m="3990980">independently--</span> <span m="3992010">in</span> <span m="3992130">fact,</span> <span m="3992410">mutually</span> <span m="3992900">independently--</span> <span m="3999070">with</span> <span m="3999230">probability</span> <span m="3999810">p.</span> <span m="4002790">And</span> <span m="4003120">p will</span> <span m="4003350">be</span> <span m="4003470">between</span> <span m="4003850">0</span> <span m="4004200">and</span> <span m="4004330">1.</span> </p>
<p><span m="4007450">And</span> <span m="4007610">now</span> <span m="4007960">we're</span> <span m="4008200">interested</span> <span m="4009200">in</span> <span m="4009460">the</span> <span m="4009650">number</span> <span m="4010670">of</span> <span m="4010800">failures,</span> <span m="4011400">so</span> <span m="4011750">we'll</span> <span m="4012350">make</span> <span m="4012640">R</span> <span m="4012960">be a</span> <span m="4013280">random</span> <span m="4013630">variable</span> <span m="4014130">that</span> <span m="4014250">tells</span> <span m="4014720">us</span> <span m="4015610">the</span> <span m="4015700">number</span> <span m="4016100">of</span> <span m="4016450">failed</span> <span m="4016860">components.</span> <span m="4022810">And</span> <span m="4022910">it</span> <span m="4022980">turns</span> <span m="4023270">out</span> <span m="4023420">the</span> <span m="4023580">answer</span> <span m="4024930">is</span> <span m="4025440">simply</span> <span m="4025750">that</span> <span m="4025950">function,</span> <span m="4027210">which</span> <span m="4027350">we'll</span> <span m="4027470">prove</span> <span m="4027820">is</span> <span m="4027960">a</span> <span m="4028020">theorem.</span> <span m="4029420">The</span> <span m="4029520">probability</span> <span m="4030060">that</span> <span m="4030200">R</span> <span m="4030600">equals</span> <span m="4031000">k</span> <span m="4032570">is</span> <span m="4032680">simply</span> <span m="4034510">fnp</span> <span m="4036074">of</span> <span m="4036476">k</span> <span m="4038700">[INAUDIBLE].</span> <span m="4043870">So</span> <span m="4043920">the</span> <span m="4044080">general</span> <span m="4044460">binomial</span> <span m="4044950">distribution</span> <span m="4046310">gives</span> <span m="4046620">you</span> <span m="4046720">the</span> <span m="4046840">distribution</span> <span m="4047580">on</span> <span m="4047690">the</span> <span m="4047770">number</span> <span m="4048230">of</span> <span m="4048320">failures</span> <span m="4049850">in</span> <span m="4050050">a</span> <span m="4050110">system</span> <span m="4050510">with</span> <span m="4050600">n</span> <span m="4050760">components</span> <span m="4051940">where</span> <span m="4052060">they</span> <span m="4052250">fail</span> <span m="4052730">with</span> <span m="4052790">probability</span> <span m="4053220">p.</span> <span m="4054638">So</span> <span m="4055132">let's prove</span> <span m="4055626">that.</span> </p>
<p><span m="4067990">To</span> <span m="4068370">do</span> <span m="4068530">that,</span> <span m="4068750">we're</span> <span m="4068830">going</span> <span m="4068920">to</span> <span m="4068970">construct</span> <span m="4069490">our</span> <span m="4070140">tree</span> <span m="4070720">again.</span> <span m="4071920">It's a</span> <span m="4071970">little</span> <span m="4072200">big</span> <span m="4072480">because</span> <span m="4073020">you've</span> <span m="4073150">got</span> <span m="4073310">n</span> <span m="4073480">components.</span> <span m="4075000">But</span> <span m="4075130">you</span> <span m="4075230">look</span> <span m="4075380">at</span> <span m="4075440">the</span> <span m="4075510">first</span> <span m="4075820">component,</span> <span m="4077180">and</span> <span m="4077430">that</span> <span m="4077600">can</span> <span m="4077720">fail</span> <span m="4078530">or</span> <span m="4078700">not.</span> <span m="4080480">And</span> <span m="4080900">it</span> <span m="4081060">fails</span> <span m="4081500">a</span> <span m="4081560">probability</span> <span m="4082020">p.</span> <span m="4082430">It's</span> <span m="4083070">OK</span> <span m="4083440">with</span> <span m="4083570">probability</span> <span m="4083930">1</span> <span m="4084160">minus</span> <span m="4084510">p.</span> <span m="4085710">And</span> <span m="4085840">you</span> <span m="4085900">have</span> <span m="4086060">the</span> <span m="4086150">second</span> <span m="4086580">component.</span> <span m="4088132">It</span> <span m="4088510">can</span> <span m="4088650">fail</span> <span m="4089380">or</span> <span m="4089510">not.</span> <span m="4090740">Again</span> <span m="4091050">p</span> <span m="4091310">and</span> <span m="4091570">1</span> <span m="4091750">minus</span> <span m="4092130">p.</span> <span m="4100210">And</span> <span m="4100390">you</span> <span m="4100460">keep</span> <span m="4100689">on</span> <span m="4100859">going</span> <span m="4101219">until</span> <span m="4101580">you</span> <span m="4101689">get</span> <span m="4101850">to</span> <span m="4101910">the</span> <span m="4102109">nth</span> <span m="4102380">component.</span> <span m="4107229">And</span> <span m="4107420">that</span> <span m="4107590">can</span> <span m="4107710">fail</span> <span m="4108649">or</span> <span m="4108800">not,</span> <span m="4109800">the</span> <span m="4109910">probability</span> <span m="4110319">p</span> <span m="4110680">or</span> <span m="4110760">1</span> <span m="4110880">minus</span> <span m="4111180">p,</span> <span m="4112850">in</span> <span m="4112910">general</span> <span m="4113390">down</span> <span m="4113630">here.</span> </p>
<p><span m="4120140">And</span> <span m="4120240">now</span> <span m="4120470">we</span> <span m="4120580">look</span> <span m="4120750">at</span> <span m="4120790">all</span> <span m="4120950">the</span> <span m="4121050">sample</span> <span m="4121500">points</span> <span m="4122300">out</span> <span m="4122470">here.</span> <span m="4127120">And</span> <span m="4127319">it's</span> <span m="4127470">all</span> <span m="4128250">length</span> <span m="4128729">and</span> <span m="4129470">vectors</span> <span m="4131189">of</span> <span m="4131319">failure</span> <span m="4132109">or</span> <span m="4132250">not.</span> <span m="4134160">So</span> <span m="4134300">the</span> <span m="4134450">top</span> <span m="4135770">sample</span> <span m="4136160">point</span> <span m="4136470">here</span> <span m="4136779">would</span> <span m="4136920">be</span> <span m="4137890">n</span> <span m="4138370">good</span> <span m="4138609">components.</span> <span m="4141750">All</span> <span m="4141819">right,</span> <span m="4142100">the</span> <span m="4142189">next</span> <span m="4142500">one</span> <span m="4142670">would</span> <span m="4142819">be</span> <span m="4143819">the</span> <span m="4143910">first</span> <span m="4144229">n</span> <span m="4144340">minus</span> <span m="4144605">1</span> <span m="4144870">are</span> <span m="4144979">good,</span> <span m="4145370">the</span> <span m="4145450">last</span> <span m="4145790">one</span> <span m="4145920">failed.</span> <span m="4148420">All</span> <span m="4148660">the</span> <span m="4148740">way</span> <span m="4148870">down</span> <span m="4149220">to</span> <span m="4149300">the</span> <span m="4149350">very</span> <span m="4149560">bottom,</span> <span m="4150170">you</span> <span m="4150270">can</span> <span m="4150370">have</span> <span m="4150410">all</span> <span m="4150730">n</span> <span m="4151050">fail.</span> </p>
<p><span m="4155170">All</span> <span m="4155279">right,</span> <span m="4155430">so</span> <span m="4155510">how</span> <span m="4155670">many</span> <span m="4155859">sample</span> <span m="4156260">points</span> <span m="4156590">are</span> <span m="4156710">there</span> <span m="4156880">in</span> <span m="4156939">the</span> <span m="4157020">sample</span> <span m="4157370">space</span> <span m="4160029">with</span> <span m="4160240">n</span> <span m="4160460">components?</span> <span m="4163170">2</span> <span m="4163430">to the</span> <span m="4163609">n,</span> <span m="4164395">all right?</span> <span m="4164830">Because</span> <span m="4165470">you've</span> <span m="4165560">got</span> <span m="4165710">n</span> <span m="4165880">positions.</span> <span m="4166760">There's</span> <span m="4166920">two</span> <span m="4167109">choices</span> <span m="4167680">for</span> <span m="4167870">each</span> <span m="4168350">value</span> <span m="4168700">there.</span> <span m="4172160">Now</span> <span m="4172689">how</span> <span m="4173100">many</span> <span m="4173479">of</span> <span m="4173550">the</span> <span m="4173640">sample</span> <span m="4174149">points</span> <span m="4175010">have</span> <span m="4175210">exactly</span> <span m="4176359">k</span> <span m="4177260">failed</span> <span m="4178130">components?</span> <span m="4180109">Out</span> <span m="4180250">of</span> <span m="4180290">the 2</span> <span m="4180620">to the</span> <span m="4180800">n,</span> <span m="4181300">how</span> <span m="4181600">many</span> <span m="4182670">correspond</span> <span m="4183210">to</span> <span m="4183290">k</span> <span m="4183670">of</span> <span m="4183740">the</span> <span m="4183830">components</span> <span m="4184410">fail?</span> <span m="4188779">n</span> <span m="4189090">choose</span> <span m="4189439">k--</span> <span m="4189790">now</span> <span m="4189920">this</span> <span m="4190050">goes</span> <span m="4190250">back</span> <span m="4190439">to</span> <span m="4190640">counting.</span> <span m="4191820">Remember</span> <span m="4193020">the</span> <span m="4194450">binomial</span> <span m="4194960">coefficient.</span> </p>
<p><span m="4195690">So</span> <span m="4196490">there</span> <span m="4196770">are</span> <span m="4197860">n</span> <span m="4198090">choose</span> <span m="4198460">k</span> <span m="4199840">sample</span> <span m="4200270">points</span> <span m="4205350">have</span> <span m="4206960">k</span> <span m="4207600">failed</span> <span m="4208060">components.</span> <span m="4216350">All right,</span> <span m="4216500">what's</span> <span m="4216880">the</span> <span m="4216990">probability</span> <span m="4218110">of</span> <span m="4218240">a</span> <span m="4218300">sample</span> <span m="4218770">point</span> <span m="4220060">with</span> <span m="4220270">k</span> <span m="4220740">failed</span> <span m="4221170">components?</span> <span m="4224184">Well,</span> <span m="4227420">I</span> <span m="4227670">took</span> <span m="4228130">k</span> <span m="4228520">branches</span> <span m="4229550">with</span> <span m="4229790">a</span> <span m="4229960">failure.</span> <span m="4230950">Each</span> <span m="4231190">of</span> <span m="4231270">those</span> <span m="4232770">gives</span> <span m="4232990">me</span> <span m="4233140">a p,</span> <span m="4235540">p to</span> <span m="4236020">the</span> <span m="4236160">k.</span> <span m="4236480">And</span> <span m="4236620">I</span> <span m="4236700">took</span> <span m="4237360">n</span> <span m="4237590">minus</span> <span m="4238090">k</span> <span m="4238300">branches</span> <span m="4238910">with</span> <span m="4239010">no</span> <span m="4239250">failure.</span> <span m="4240900">Each</span> <span m="4241100">one</span> <span m="4241280">of</span> <span m="4241330">those</span> <span m="4241650">multiplies</span> <span m="4242330">by a</span> <span m="4242520">1</span> <span m="4242770">minus</span> <span m="4243110">p.</span> <span m="4246370">So</span> <span m="4246620">no</span> <span m="4246700">matter</span> <span m="4247140">how</span> <span m="4247360">I</span> <span m="4247470">got</span> <span m="4247690">my</span> <span m="4247820">k</span> <span m="4247940">failures,</span> <span m="4250080">the</span> <span m="4250190">probability</span> <span m="4250870">for a</span> <span m="4251020">particular</span> <span m="4251580">sample</span> <span m="4251980">point</span> <span m="4252280">with</span> <span m="4252420">k</span> <span m="4252650">failures</span> <span m="4254000">is</span> <span m="4254190">just</span> <span m="4254470">that,</span> <span m="4255550">because</span> <span m="4255680">I</span> <span m="4255770">had</span> <span m="4256350">p</span> <span m="4256930">get</span> <span m="4257110">factored</span> <span m="4257450">in</span> <span m="4257590">k</span> <span m="4257830">times,</span> <span m="4258170">in</span> <span m="4258510">the</span> <span m="4258660">failures,</span> <span m="4259550">1</span> <span m="4259800">minus</span> <span m="4260170">p</span> <span m="4260380">get</span> <span m="4260550">factored</span> <span m="4260900">in</span> <span m="4261070">n</span> <span m="4261250">minus</span> <span m="4261580">k</span> <span m="4261730">times</span> <span m="4262140">for</span> <span m="4262230">the</span> <span m="4263240">situations</span> <span m="4263840">where it</span> <span m="4263990">worked</span> <span m="4264240">out</span> <span m="4264340">all</span> <span m="4264410">right.</span> </p>
<p><span m="4267970">So</span> <span m="4268050">I've</span> <span m="4268160">got</span> <span m="4268380">this</span> <span m="4268650">many</span> <span m="4268860">sample</span> <span m="4269310">points.</span> <span m="4270980">Each</span> <span m="4271230">of</span> <span m="4271330">them</span> <span m="4274700">has</span> <span m="4275030">probability</span> <span m="4280440">p</span> <span m="4280660">to</span> <span m="4280730">the</span> <span m="4280880">k</span> <span m="4281490">times</span> <span m="4282580">1</span> <span m="4282870">minus</span> <span m="4283220">p</span> <span m="4284220">n</span> <span m="4284620">minus</span> <span m="4284960">k.</span> <span m="4288230">Any</span> <span m="4288380">questions</span> <span m="4288810">about</span> <span m="4289050">that,</span> <span m="4289490">why</span> <span m="4289680">that's</span> <span m="4289920">the</span> <span m="4290000">case?</span> <span m="4293700">All</span> <span m="4293820">right,</span> <span m="4294000">so</span> <span m="4294110">now</span> <span m="4294340">I</span> <span m="4294410">can</span> <span m="4294570">compute</span> <span m="4294950">the</span> <span m="4295040">probability</span> <span m="4295650">there</span> <span m="4295860">are</span> <span m="4296030">k</span> <span m="4296320">failures.</span> <span m="4298730">The</span> <span m="4298820">probability</span> <span m="4299290">that</span> <span m="4299410">R</span> <span m="4299780">equals</span> <span m="4300190">k</span> <span m="4302210">is</span> <span m="4302400">simply</span> <span m="4302840">n</span> <span m="4303040">choose</span> <span m="4303370">k,</span> <span m="4305400">number</span> <span m="4305640">of</span> <span m="4305710">sample</span> <span m="4306020">points</span> <span m="4306440">times</span> <span m="4306800">their</span> <span m="4307010">probability,</span> <span m="4307870">since they're</span> <span m="4307980">all</span> <span m="4308090">the</span> <span m="4308180">same</span> <span m="4309130">for</span> <span m="4309700">k</span> <span m="4309940">failures.</span> <span m="4311920">And</span> <span m="4311980">that,</span> <span m="4312270">of</span> <span m="4312370">course,</span> <span m="4313530">is</span> <span m="4313600">just</span> <span m="4313870">the</span> <span m="4313950">formula</span> <span m="4315830">for</span> <span m="4315910">the</span> <span m="4316040">general</span> <span m="4316400">binomial</span> <span m="4316870">distribution.</span> <span m="4319290">So</span> <span m="4319430">that</span> <span m="4319720">equals</span> <span m="4320850">fnp</span> <span m="4322344">of</span> <span m="4322776">k,</span> <span m="4323820">which</span> <span m="4324030">is</span> <span m="4324120">what</span> <span m="4324330">we</span> <span m="4324750">are</span> <span m="4325160">trying</span> <span m="4325653">to prove.</span> <span m="4331570">OK.</span> <span m="4334050">Any</span> <span m="4334210">questions</span> <span m="4334750">about</span> <span m="4334970">that?</span> </p>
<p><span m="4336840">So</span> <span m="4336940">that's</span> <span m="4337060">why</span> <span m="4337380">it's</span> <span m="4337640">important,</span> <span m="4338210">because</span> <span m="4338830">there's</span> <span m="4338980">a</span> <span m="4339020">lot</span> <span m="4339260">of</span> <span m="4339320">situations</span> <span m="4340130">where</span> <span m="4340970">you're</span> <span m="4341120">interested</span> <span m="4341580">in</span> <span m="4341650">the</span> <span m="4341730">number</span> <span m="4342100">of--</span> <span m="4342570">if</span> <span m="4342680">you</span> <span m="4342740">had</span> <span m="4342900">n</span> <span m="4343140">possible</span> <span m="4343630">things,</span> <span m="4344330">what's</span> <span m="4344520">the</span> <span m="4344600">chance</span> <span m="4344940">k of them</span> <span m="4345360">happen?</span> <span m="4346590">And</span> <span m="4346920">it's</span> <span m="4347840">just</span> <span m="4347980">given</span> <span m="4348200">by</span> <span m="4348320">this</span> <span m="4348500">formula.</span> <span m="4350630">Now</span> <span m="4352420">you</span> <span m="4352580">can</span> <span m="4352750">calculate</span> <span m="4353440">this,</span> <span m="4354430">but,</span> <span m="4355870">if</span> <span m="4355980">I</span> <span m="4356060">just</span> <span m="4356300">told</span> <span m="4356610">you,</span> <span m="4356850">for</span> <span m="4356910">example,</span> <span m="4358760">maybe</span> <span m="4359020">I'm</span> <span m="4359130">looking</span> <span m="4359460">at</span> <span m="4360430">n</span> <span m="4360690">is</span> <span m="4360810">100</span> <span m="4361560">and</span> <span m="4362230">even</span> <span m="4362470">p</span> <span m="4362760">is</span> <span m="4362890">1/2,</span> <span m="4365310">you</span> <span m="4365460">know,</span> <span m="4365770">what's</span> <span m="4366220">the</span> <span m="4366330">probability</span> <span m="4367110">that</span> <span m="4367290">we'd</span> <span m="4367490">get</span> <span m="4369540">25--</span> <span m="4370370">say,</span> <span m="4371030">what's the</span> <span m="4371200">probably</span> <span m="4371460">if getting</span> <span m="4371690">50</span> <span m="4372120">failed</span> <span m="4372430">components?</span> </p>
<p><span m="4374140">Or</span> <span m="4374410">if</span> <span m="4374470">I</span> <span m="4374550">flip</span> <span m="4374800">100</span> <span m="4375140">coins</span> <span m="4377140">and</span> <span m="4377340">they're</span> <span m="4377470">fair.</span> <span m="4378480">So</span> <span m="4378950">probability</span> <span m="4379330">half</span> <span m="4379600">of</span> <span m="4379690">getting</span> <span m="4379940">a</span> <span m="4379960">heads.</span> <span m="4380730">What's</span> <span m="4381020">the</span> <span m="4381120">chance</span> <span m="4381410">I</span> <span m="4381520">actually</span> <span m="4381900">get</span> <span m="4382160">50</span> <span m="4382470">heads</span> <span m="4383660">in</span> <span m="4383760">100</span> <span m="4384060">coins?</span> <span m="4385680">Looking</span> <span m="4386060">at</span> <span m="4386130">that,</span> <span m="4386810">it's</span> <span m="4386990">not</span> <span m="4387190">so</span> <span m="4387570">clear</span> <span m="4388110">what</span> <span m="4388290">the</span> <span m="4388440">answer</span> <span m="4388830">is.</span> <span m="4390950">In</span> <span m="4390990">fact,</span> <span m="4391300">let's</span> <span m="4391430">test</span> <span m="4391790">intuition</span> <span m="4392340">here,</span> <span m="4394230">for</span> <span m="4394290">this.</span> <span m="4394570">We're going</span> <span m="4394700">to</span> <span m="4395380">take</span> <span m="4395540">a</span> <span m="4395600">little</span> <span m="4395800">vote</span> <span m="4396140">to</span> <span m="4396220">see</span> <span m="4398060">how</span> <span m="4398290">good</span> <span m="4398520">you</span> <span m="4398640">are</span> <span m="4398740">looking</span> <span m="4398990">at</span> <span m="4399060">that</span> <span m="4399280">formula,</span> <span m="4399720">or</span> <span m="4399810">what</span> <span m="4399960">your</span> <span m="4400100">intuition</span> <span m="4400590">is</span> <span m="4400750">about</span> <span m="4400980">the</span> <span m="4401070">probability</span> <span m="4401910">of</span> <span m="4403180">getting</span> <span m="4403420">exactly</span> <span m="4403990">50</span> <span m="4404380">heads.</span> <span m="4409240">It</span> <span m="4409340">could</span> <span m="4409430">be</span> <span m="4409540">between</span> <span m="4409870">1</span> <span m="4410280">and</span> <span m="4410680">1/2,</span> <span m="4411830">a half</span> <span m="4412270">and</span> <span m="4412350">a tenth,</span> <span m="4414534">a tenth</span> <span m="4414960">and</span> <span m="4415448">a hundredth,</span> <span m="4417400">a hundredth</span> <span m="4417890">and</span> <span m="4418020">one thousandth,</span> <span m="4420030">a</span> <span m="4420130">thousandth</span> <span m="4420590">and</span> <span m="4420740">a millionth,</span> <span m="4421970">and</span> <span m="4422180">0.</span> </p>
<p><span m="4424290">I</span> <span m="4424410">want</span> <span m="4424610">to</span> <span m="4424670">know</span> <span m="4424860">what's</span> <span m="4425080">the</span> <span m="4425190">probability,</span> <span m="4425950">when</span> <span m="4426050">you</span> <span m="4426140">flip</span> <span m="4426360">100</span> <span m="4426820">mutually</span> <span m="4427240">independent</span> <span m="4427690">coins,</span> <span m="4428810">you</span> <span m="4428920">get</span> <span m="4429040">exactly</span> <span m="4429640">50</span> <span m="4430020">heads?</span> <span m="4432370">How</span> <span m="4432470">many</span> <span m="4432650">people</span> <span m="4432950">think</span> <span m="4433180">it's</span> <span m="4433470">at</span> <span m="4433540">least</span> <span m="4433780">a</span> <span m="4433820">half,</span> <span m="4434870">that</span> <span m="4435020">you</span> <span m="4435110">get</span> <span m="4435240">half</span> <span m="4435470">heads?</span> <span m="4437900">Nobody</span> <span m="4438230">likes</span> <span m="4438490">that.</span> <span m="4438920">How</span> <span m="4439050">many</span> <span m="4439220">people</span> <span m="4439470">think</span> <span m="4439600">it's</span> <span m="4439710">between</span> <span m="4439970">a half</span> <span m="4440250">and</span> <span m="4440330">a tenth?</span> <span m="4442640">One</span> <span m="4443150">vote.</span> <span m="4443890">A tenth</span> <span m="4444260">and</span> <span m="4444760">a  hundredth?</span> <span m="4446520">More.</span> <span m="4449400">Doesn't</span> <span m="4449680">it have</span> <span m="4449840">to</span> <span m="4449910">be</span> <span m="4449990">at</span> <span m="4450060">least</span> <span m="4450350">a hundredth?</span> <span m="4450730">That's</span> <span m="4450980">sort of</span> <span m="4451270">the</span> <span m="4451320">most</span> <span m="4451570">likely</span> <span m="4451870">outcome.</span> <span m="4452250">There's</span> <span m="4452400">only--</span> <span m="4452730">how</span> <span m="4452890">many</span> <span m="4453050">think</span> <span m="4453270">a hundredth</span> <span m="4453730">and</span> <span m="4453810">a thousandth?</span> <span m="4456600">All right.</span> <span m="4457350">A thousandth</span> <span m="4457850">and a</span> <span m="4457910">millionth?</span> <span m="4459840">Whoa,</span> <span m="4460850">so</span> <span m="4461020">you've</span> <span m="4461260">got to</span> <span m="4461280">believe it</span> <span m="4461490">is</span> <span m="4461660">not</span> <span m="4462040">likely</span> <span m="4462340">to</span> <span m="4462400">get</span> <span m="4462530">exactly</span> <span m="4463050">50</span> <span m="4463370">heads</span> <span m="4463620">when</span> <span m="4463720">I</span> <span m="4463770">flip</span> <span m="4463960">100</span> <span m="4464270">coins.</span> </p>
<p><span m="4465560">All right,</span> <span m="4465810">well</span> <span m="4465920">the</span> <span m="4466110">answer</span> <span m="4466470">is</span> <span m="4467060">actually</span> <span m="4468690">between</span> <span m="4469150">a</span> <span m="4469230">tenth</span> <span m="4469560">and</span> <span m="4469620">a hundredth</span> <span m="4470390">here.</span> <span m="4473730">It's</span> <span m="4473830">about</span> <span m="4474060">8%.</span> <span m="4474910">And</span> <span m="4475030">we're</span> <span m="4475130">going</span> <span m="4475220">to</span> <span m="4475280">compute</span> <span m="4475680">that.</span> <span m="4476570">But</span> <span m="4476710">here's</span> <span m="4476950">another</span> <span m="4477270">one.</span> <span m="4478330">I</span> <span m="4478420">flip</span> <span m="4478650">100</span> <span m="4478980">coins.</span> <span m="4480370">What's</span> <span m="4480870">the</span> <span m="4480980">probability</span> <span m="4482220">of</span> <span m="4482420">getting</span> <span m="4483300">at</span> <span m="4483530">most</span> <span m="4484570">25</span> <span m="4485240">heads?</span> <span m="4486520">So</span> <span m="4486680">here,</span> <span m="4487450">doesn't</span> <span m="4487470">have to be</span> <span m="4487710">exactly</span> <span m="4488150">25.</span> <span m="4488605">It could be</span> <span m="4489060">25,</span> <span m="4489515">24,</span> <span m="4489970">23,</span> <span m="4490425">22,</span> <span m="4491340">all the</span> <span m="4491560">way</span> <span m="4491690">down</span> <span m="4491910">to</span> <span m="4491980">no</span> <span m="4492210">heads.</span> <span m="4493400">So</span> <span m="4493500">a</span> <span m="4493580">lot</span> <span m="4493860">of</span> <span m="4493920">chances,</span> <span m="4495140">all</span> <span m="4495220">right?</span> <span m="4495560">To</span> <span m="4495690">get</span> <span m="4495940">at</span> <span m="4496040">most</span> <span m="4496260">25</span> <span m="4496690">heads.</span> <span m="4497630">How</span> <span m="4497680">many</span> <span m="4497850">people</span> <span m="4498110">think</span> <span m="4498290">it's</span> <span m="4498440">here?</span> <span m="4500200">Some,</span> <span m="4500340">yeah,</span> <span m="4500650">you've got</span> <span m="4500960">a</span> <span m="4500980">lot</span> <span m="4501150">of</span> <span m="4501200">chances.</span> <span m="4502770">How</span> <span m="4502930">about</span> <span m="4503030">here?</span> <span m="4504885">Yeah,</span> <span m="4505360">OK.</span> <span m="4506230">What</span> <span m="4506400">about</span> <span m="4507160">between</span> <span m="4507540">a tenth</span> <span m="4507730">and</span> <span m="4507810">a hundredth?</span> <span m="4509858">One</span> <span m="4510350">person,</span> <span m="4510780">two</span> <span m="4510940">people</span> <span m="4511250">left.</span> <span m="4512940">Between</span> <span m="4513220">a hundredth</span> <span m="4513630">and a</span> <span m="4513720">thousandth?</span> <span m="4515930">Nobody</span> <span m="4516270">left?</span> <span m="4516620">Anybody</span> <span m="4516990">here?</span> <span m="4518050">Thousandth</span> <span m="4518470">and a</span> <span m="4518590">millionth?</span> <span m="4520630">One</span> <span m="4520915">person</span> <span m="4521200">is</span> <span m="4521500">left, less</span> <span m="4521750">than</span> <span m="4521870">a</span> <span m="4521920">million.</span> <span m="4522340">1</span> <span m="4522590">in a</span> <span m="4522630">million?</span> <span m="4524630">There's</span> <span m="4524850">the</span> <span m="4524950">contrarian.</span> </p>
<p><span m="4525880">And</span> <span m="4526090">you're</span> <span m="4526260">right,</span> <span m="4528090">the</span> <span m="4528190">chance</span> <span m="4528770">of</span> <span m="4528870">getting</span> <span m="4529800">25</span> <span m="4530370">or</span> <span m="4530430">less</span> <span m="4530690">heads</span> <span m="4531310">is</span> <span m="4531610">less</span> <span m="4532000">than</span> <span m="4532140">1</span> <span m="4532550">in</span> <span m="4532700">a</span> <span m="4532740">million.</span> <span m="4534330">So if</span> <span m="4534490">somebody</span> <span m="4534870">does</span> <span m="4535190">it,</span> <span m="4536470">his</span> <span m="4536620">name</span> <span m="4536950">better</span> <span m="4537170">be</span> <span m="4537320">Persi</span> <span m="4537620">Diaconis.</span> <span m="4539020">To</span> <span m="4539500">get</span> <span m="4539640">it</span> <span m="4539800">consistently</span> <span m="4540115">to</span> <span m="4540430">70,</span> <span m="4540800">[INAUDIBLE]</span> <span m="4541650">25</span> <span m="4542090">or</span> <span m="4542170">fewer</span> <span m="4542430">heads.</span> <span m="4542710">You get</span> <span m="4542860">75</span> <span m="4543320">tails</span> <span m="4543650">or</span> <span m="4543700">more</span> <span m="4544550">is</span> <span m="4544700">extremely</span> <span m="4545820">unlikely,</span> <span m="4547830">all</span> <span m="4547930">right?</span> <span m="4548160">So</span> <span m="4548260">to</span> <span m="4548410">see</span> <span m="4548690">why</span> <span m="4548940">that's</span> <span m="4549210">the</span> <span m="4549310">case,</span> <span m="4550510">we've</span> <span m="4550660">got</span> <span m="4550780">to</span> <span m="4550830">do</span> <span m="4550930">a</span> <span m="4550990">little</span> <span m="4551240">work</span> <span m="4551990">on</span> <span m="4552200">this</span> <span m="4552390">formula.</span> <span m="4553900">Because</span> <span m="4554670">we're</span> <span m="4554800">saying</span> <span m="4555240">that</span> <span m="4555360">if</span> <span m="4555510">we</span> <span m="4555600">sum</span> <span m="4555950">this</span> <span m="4556190">up</span> <span m="4556470">for</span> <span m="4556760">k</span> <span m="4557090">ending</span> <span m="4557450">100</span> <span m="4557820">and k</span> <span m="4558100">being</span> <span m="4558890">0</span> <span m="4559080">to</span> <span m="4559160">25,</span> <span m="4559880">and</span> <span m="4560020">p</span> <span m="4560270">being</span> <span m="4560550">a</span> <span m="4560570">half,</span> <span m="4561470">it's</span> <span m="4561670">an</span> <span m="4561750">incredibly</span> <span m="4562410">tiny</span> <span m="4562790">number.</span> <span m="4564600">So</span> <span m="4564660">let's</span> <span m="4564790">see</span> <span m="4564900">why</span> <span m="4565050">that</span> <span m="4565270">is.</span> </p>
<p><span m="4566020">And</span> <span m="4566130">this</span> <span m="4566290">phenomenon</span> <span m="4566950">is</span> <span m="4567130">course</span> <span m="4567330">going to</span> <span m="4567430">be</span> <span m="4567510">important in</span> <span m="4567920">computer</span> <span m="4568270">science</span> <span m="4569690">because</span> <span m="4569890">it's</span> <span m="4570050">going</span> <span m="4570150">to</span> <span m="4570190">enable</span> <span m="4570560">you</span> <span m="4570700">to</span> <span m="4570820">tell</span> <span m="4571240">almost</span> <span m="4571560">exactly</span> <span m="4572140">how</span> <span m="4572290">many</span> <span m="4572480">components</span> <span m="4573010">are</span> <span m="4573070">going</span> <span m="4573180">to</span> <span m="4573250">fail</span> <span m="4574630">in</span> <span m="4574750">a</span> <span m="4574820">system,</span> <span m="4575120">if they're</span> <span m="4575619">mutually</span> <span m="4576118">independent.</span> <span m="4591100">All right,</span> <span m="4591200">now</span> <span m="4591410">I'm</span> <span m="4591600">not</span> <span m="4591770">going</span> <span m="4591860">to</span> <span m="4591930">drag</span> <span m="4592370">you</span> <span m="4592460">through</span> <span m="4592580">the</span> <span m="4592690">math.</span> <span m="4593040">It's</span> <span m="4593190">bad</span> <span m="4593410">enough</span> <span m="4593640">I'm</span> <span m="4593740">going</span> <span m="4593840">to</span> <span m="4593930">write</span> <span m="4594120">it</span> <span m="4594190">on</span> <span m="4594290">the</span> <span m="4594360">board.</span> <span m="4596760">There's</span> <span m="4597280">a</span> <span m="4597350">bunch</span> <span m="4597620">of</span> <span m="4597670">this</span> <span m="4597840">in</span> <span m="4597920">the</span> <span m="4598000">text.</span> <span m="4600320">And</span> <span m="4600470">instead</span> <span m="4600720">of</span> <span m="4600840">k,</span> <span m="4601260">I'm</span> <span m="4601340">going</span> <span m="4601460">to</span> <span m="4601520">represent</span> <span m="4602170">by</span> <span m="4602420">a</span> <span m="4602500">parameter</span> <span m="4603100">alpha</span> <span m="4603560">times</span> <span m="4603940">n</span> <span m="4604510">to</span> <span m="4604630">be</span> <span m="4604770">the</span> <span m="4604960">integer</span> <span m="4605260">k.</span> <span m="4606452">All right,</span> <span m="4606850">this</span> <span m="4607030">will</span> <span m="4607100">replace</span> <span m="4607550">k with</span> <span m="4607970">a</span> <span m="4608110">new</span> <span m="4608280">parameter</span> <span m="4608820">alpha.</span> </p>
<p><span m="4610330">It's</span> <span m="4610560">at</span> <span m="4610750">most,</span> <span m="4611660">and</span> <span m="4611810">also</span> <span m="4612180">it's</span> <span m="4612400">asymptotically</span> <span m="4613600">equivalent</span> <span m="4614190">using</span> <span m="4614480">tilde--</span> <span m="4615210">both</span> <span m="4615430">of</span> <span m="4615510">those</span> <span m="4615690">are</span> <span m="4615760">true--</span> <span m="4617070">to</span> <span m="4617250">this</span> <span m="4617600">nasty</span> <span m="4618090">looking</span> <span m="4618390">expression,</span> <span m="4620570">2</span> <span m="4620920">to</span> <span m="4621270">the</span> <span m="4622330">alpha</span> <span m="4622940">log</span> <span m="4625200">p</span> <span m="4625440">over</span> <span m="4625670">alpha</span> <span m="4628900">plus</span> <span m="4629970">1</span> <span m="4630300">minus</span> <span m="4630750">alpha</span> <span m="4632900">log</span> <span m="4633560">1</span> <span m="4633840">minus</span> <span m="4634220">p</span> <span m="4634770">over</span> <span m="4635070">1</span> <span m="4635290">minus</span> <span m="4635650">alpha</span> <span m="4638180">times</span> <span m="4638740">and</span> <span m="4639120">n--</span> <span m="4639530">that's</span> <span m="4639700">a</span> <span m="4639740">big</span> <span m="4639990">number,</span> <span m="4640450">it's 100--</span> <span m="4642150">over</span> <span m="4644630">square</span> <span m="4645180">root</span> <span m="4645760">2</span> <span m="4646000">pi</span> <span m="4647440">alpha</span> <span m="4648240">1</span> <span m="4648470">minus</span> <span m="4648830">alpha</span> <span m="4650120">n.</span> <span m="4651840">And</span> <span m="4651940">this,</span> <span m="4652160">of course,</span> <span m="4652400">alpha</span> <span m="4652690">is</span> <span m="4652760">between</span> <span m="4653160">0</span> <span m="4653480">and</span> <span m="4653590">1.</span> </p>
<p><span m="4657790">Now</span> <span m="4658300">when</span> <span m="4658420">you</span> <span m="4658520">do</span> <span m="4658660">the</span> <span m="4658790">derivative</span> <span m="4659260">on</span> <span m="4659410">this</span> <span m="4659610">thing,</span> <span m="4659970">it</span> <span m="4660070">turns</span> <span m="4660230">out</span> <span m="4660340">its</span> <span m="4660470">maximum</span> <span m="4661030">value</span> <span m="4661490">is</span> <span m="4661610">when</span> <span m="4661780">alpha</span> <span m="4662070">equals</span> <span m="4662460">p.</span> <span m="4671320">And</span> <span m="4671660">when</span> <span m="4671770">you</span> <span m="4671840">have</span> <span m="4672080">alpha</span> <span m="4672470">equals</span> <span m="4672890">p,</span> <span m="4674740">you</span> <span m="4674800">get</span> <span m="4675130">log</span> <span m="4675430">of</span> <span m="4675490">1</span> <span m="4675830">is</span> <span m="4675990">0,</span> <span m="4677580">log</span> <span m="4677810">of</span> <span m="4677890">1</span> <span m="4678110">is</span> <span m="4678260">0.</span> <span m="4678640">All</span> <span m="4679030">this</span> <span m="4679750">messy</span> <span m="4680110">stuff</span> <span m="4680350">goes</span> <span m="4680550">away</span> <span m="4680870">and we</span> <span m="4681020">just</span> <span m="4681310">get</span> <span m="4681480">that.</span> <span m="4683090">And</span> <span m="4683260">so</span> <span m="4683380">in</span> <span m="4683460">that</span> <span m="4683680">case,</span> <span m="4685200">fnp</span> <span m="4687100">of</span> <span m="4687220">pn</span> <span m="4688516">is</span> <span m="4688950">at</span> <span m="4689210">most,</span> <span m="4689860">and</span> <span m="4690030">also</span> <span m="4690520">asymptotically</span> <span m="4691300">equal</span> <span m="4691580">to</span> <span m="4692660">1</span> <span m="4693090">over</span> <span m="4694230">square</span> <span m="4694690">root</span> <span m="4694940">2</span> <span m="4695140">pi</span> <span m="4696700">p</span> <span m="4697090">1</span> <span m="4697480">minus</span> <span m="4697810">p</span> <span m="4698690">n.</span> </p>
<p><span m="4701650">And</span> <span m="4701780">now</span> <span m="4702010">you</span> <span m="4702160">can</span> <span m="4702300">plug in</span> <span m="4702660">values</span> <span m="4703070">and</span> <span m="4703180">compute</span> <span m="4703560">things.</span> <span m="4703860">For</span> <span m="4703950">example,</span> <span m="4704370">if</span> <span m="4704470">I</span> <span m="4704530">flip</span> <span m="4704760">100</span> <span m="4705120">coins,</span> <span m="4706880">and</span> <span m="4707060">I</span> <span m="4707100">want</span> <span m="4707310">to</span> <span m="4707350">know</span> <span m="4707440">the</span> <span m="4707540">probability</span> <span m="4708170">of</span> <span m="4708250">getting</span> <span m="4708600">50--</span> <span m="4709720">that</span> <span m="4709870">means</span> <span m="4710080">alpha's</span> <span m="4710400">a</span> <span m="4710450">half,</span> <span m="4711560">and</span> <span m="4711750">they're</span> <span m="4711870">fair,</span> <span m="4712240">which</span> <span m="4712410">means</span> <span m="4712620">p</span> <span m="4712850">is</span> <span m="4712990">a</span> <span m="4713040">half,</span> <span m="4715670">the</span> <span m="4715790">answer</span> <span m="4716040">is</span> <span m="4716140">8%.</span> <span m="4718080">So</span> <span m="4718270">n</span> <span m="4718480">equals</span> <span m="4718730">100</span> <span m="4719170">coins,</span> <span m="4720210">p</span> <span m="4720480">is</span> <span m="4720630">a</span> <span m="4720680">half,</span> <span m="4722100">then</span> <span m="4722240">the</span> <span m="4722330">probability</span> <span m="4723070">of</span> <span m="4723770">50</span> <span m="4724260">heads.</span> <span m="4729370">I</span> <span m="4729480">just</span> <span m="4729680">plug in</span> <span m="4730070">p</span> <span m="4730310">is</span> <span m="4730460">a</span> <span m="4730510">half</span> <span m="4730810">here,</span> <span m="4731120">I</span> <span m="4731180">get</span> <span m="4732610">1</span> <span m="4732880">over</span> <span m="4733160">square</span> <span m="4733520">root</span> <span m="4733770">50</span> <span m="4734070">pi,</span> <span m="4735840">which</span> <span m="4735960">equals</span> <span m="4736340">0.080</span> <span m="4738620">and</span> <span m="4738990">so</span> <span m="4739170">forth.</span> </p>
<p><span m="4740670">So</span> <span m="4740690">there's</span> <span m="4740850">an</span> <span m="4740960">8%</span> <span m="4741550">chance</span> <span m="4742120">of</span> <span m="4742290">getting</span> <span m="4743050">exactly</span> <span m="4743530">50</span> <span m="4743860">heads.</span> <span m="4746040">Now</span> <span m="4746230">let's</span> <span m="4746450">look</span> <span m="4746570">at</span> <span m="4746620">the</span> <span m="4746700">probability</span> <span m="4747280">of</span> <span m="4747370">getting</span> <span m="4747690">25</span> <span m="4748480">or</span> <span m="4748570">fewer.</span> <span m="4750670">And</span> <span m="4750820">we'll</span> <span m="4750930">see</span> <span m="4751230">why</span> <span m="4751430">that</span> <span m="4751610">is</span> <span m="4751730">so</span> <span m="4752360">surprisingly</span> <span m="4753060">small.</span> <span m="4770170">OK,</span> <span m="4770720">so</span> <span m="4774280">for</span> <span m="4774780">n</span> <span m="4775210">equal</span> <span m="4775540">100</span> <span m="4777280">and</span> <span m="4777440">p</span> <span m="4777780">equal</span> <span m="4778160">a half</span> <span m="4779360">an</span> <span m="4779560">alpha</span> <span m="4779960">equal</span> <span m="4780380">a quarter,</span> <span m="4780805">because</span> <span m="4781230">we</span> <span m="4781370">want</span> <span m="4781550">25</span> <span m="4782180">heads,</span> <span m="4782630">I'll</span> <span m="4782730">get</span> <span m="4783010">exactly</span> <span m="4783470">25</span> <span m="4783950">heads</span> <span m="4784200">first.</span> <span m="4786110">Probability</span> <span m="4786600">of</span> <span m="4786670">25</span> <span m="4787310">heads</span> <span m="4790870">is</span> <span m="4791070">at</span> <span m="4791180">most,</span> <span m="4792460">well</span> <span m="4792600">that</span> <span m="4792820">square</span> <span m="4793240">root</span> <span m="4793480">thing</span> <span m="4793740">comes</span> <span m="4794070">out</span> <span m="4794360">to</span> <span m="4794510">about</span> <span m="4795010">0.09.</span> <span m="4797900">Then</span> <span m="4798050">I</span> <span m="4798110">get</span> <span m="4798410">2</span> <span m="4798650">to</span> <span m="4798730">something.</span> <span m="4800230">I</span> <span m="4800340">get</span> <span m="4800640">2 to</span> <span m="4801050">the</span> <span m="4802250">minus,</span> <span m="4802930">all</span> <span m="4803070">those</span> <span m="4803330">logs</span> <span m="4803760">come</span> <span m="4803900">out</span> <span m="4804050">the</span> <span m="4804140">0.1887</span> <span m="4808060">times</span> <span m="4809380">n,</span> <span m="4810580">which</span> <span m="4810620">is</span> <span m="4810750">the</span> <span m="4810840">kicker,</span> <span m="4811160">that's</span> <span m="4811370">100.</span> <span m="4812920">So</span> <span m="4813050">I</span> <span m="4813120">get</span> <span m="4813290">a</span> <span m="4813400">2 to</span> <span m="4813690">the</span> <span m="4813780">minus</span> <span m="4814300">18th</span> <span m="4814810">here.</span> <span m="4816730">And</span> <span m="4817870">that</span> <span m="4818070">makes</span> <span m="4818320">this</span> <span m="4818540">thing</span> <span m="4818850">be</span> <span m="4819560">smaller</span> <span m="4820100">than</span> <span m="4821080">or</span> <span m="4821290">equal</span> <span m="4821620">about</span> <span m="4821895">to</span> <span m="4822170">1.913</span> <span m="4824550">times</span> <span m="4824790">10</span> <span m="4825040">to the</span> <span m="4825120">minus</span> <span m="4825500">7.</span> <span m="4826230">So</span> <span m="4826360">about</span> <span m="4826590">1 in</span> <span m="4826920">5</span> <span m="4827310">million.</span> </p>
<p><span m="4829960">The</span> <span m="4830160">reason</span> <span m="4830540">this</span> <span m="4830740">gets</span> <span m="4830970">so</span> <span m="4831160">small</span> <span m="4832260">is</span> <span m="4832400">because</span> <span m="4832680">you</span> <span m="4832760">get</span> <span m="4832910">the</span> <span m="4833080">n</span> <span m="4833400">in</span> <span m="4833480">that</span> <span m="4833660">exponent</span> <span m="4834310">up</span> <span m="4834710">there.</span> <span m="4836010">And</span> <span m="4836110">of</span> <span m="4836170">course,</span> <span m="4836400">this</span> <span m="4836530">is</span> <span m="4836650">all</span> <span m="4836950">computed</span> <span m="4837330">using</span> <span m="4837690">Stirling's</span> <span m="4838160">formula,</span> <span m="4838940">if</span> <span m="4839090">you</span> <span m="4839190">actually</span> <span m="4839410">want</span> <span m="4839560">to</span> <span m="4839610">go</span> <span m="4839730">through</span> <span m="4839840">the</span> <span m="4839940">calculations.</span> <span m="4841370">You</span> <span m="4841480">go</span> <span m="4841620">start</span> <span m="4842080">with</span> <span m="4842840">n</span> <span m="4843070">choose</span> <span m="4843380">k,</span> <span m="4843880">which</span> <span m="4844090">is</span> <span m="4844170">your</span> <span m="4844320">n</span> <span m="4844520">factorial,</span> <span m="4846190">times</span> <span m="4846840">over</span> <span m="4847050">k</span> <span m="4847260">factorial</span> <span m="4847830">n</span> <span m="4847960">minus</span> <span m="4848220">k</span> <span m="4848370">factorial.</span> <span m="4848980">Plug in</span> <span m="4849350">Stirling's</span> <span m="4849810">formula</span> <span m="4850260">and you</span> <span m="4850360">do</span> <span m="4850540">a</span> <span m="4850630">bunch</span> <span m="4851000">of</span> <span m="4851060">messy</span> <span m="4851360">stuff,</span> <span m="4852280">and</span> <span m="4852410">these</span> <span m="4852630">things</span> <span m="4852920">pop</span> <span m="4853200">out.</span> <span m="4854840">And</span> <span m="4854900">so</span> <span m="4855020">it</span> <span m="4855090">gets</span> <span m="4855280">exponentially</span> <span m="4855860">small,</span> <span m="4856260">exponentially</span> <span m="4856770">fast.</span> </p>
<p><span m="4857720">In</span> <span m="4857730">fact,</span> <span m="4858090">if</span> <span m="4858190">you</span> <span m="4858290">were</span> <span m="4858390">to</span> <span m="4858490">plot</span> <span m="4858680">this</span> <span m="4858920">thing,</span> <span m="4860270">if</span> <span m="4860400">you</span> <span m="4860510">plot</span> <span m="4860890">the</span> <span m="4860950">binomial</span> <span m="4861430">distribution</span> <span m="4862090">function,</span> <span m="4863510">it</span> <span m="4863650">looks</span> <span m="4865450">something</span> <span m="4865910">like</span> <span m="4866110">this.</span> <span m="4877330">All</span> <span m="4877450">right,</span> <span m="4877640">so</span> <span m="4877740">I</span> <span m="4877800">have</span> <span m="4878610">0</span> <span m="4879060">to</span> <span m="4879220">n,</span> <span m="4881080">and</span> <span m="4881180">then</span> <span m="4881310">I</span> <span m="4881380">have</span> <span m="4881630">pn,</span> <span m="4883410">which</span> <span m="4883630">is</span> <span m="4883740">the</span> <span m="4883830">maximum.</span> <span m="4885300">Here's</span> <span m="4885530">the</span> <span m="4885620">probability.</span> <span m="4885915">It</span> <span m="4886210">goes</span> <span m="4886460">0</span> <span m="4886680">to</span> <span m="4886800">1.</span> <span m="4888560">And</span> <span m="4888780">your</span> <span m="4888900">maximum</span> <span m="4889390">value</span> <span m="4889770">is</span> <span m="4889920">here</span> <span m="4890630">at</span> <span m="4890750">8%</span> <span m="4891390">in</span> <span m="4891460">the</span> <span m="4891540">case</span> <span m="4891850">of</span> <span m="4892150">100</span> <span m="4892430">coins,</span> <span m="4893310">and</span> <span m="4893490">it just</span> <span m="4893780">zooms</span> <span m="4894430">down</span> <span m="4895820">exponential</span> <span m="4897380">small.</span> <span m="4899980">You</span> <span m="4900160">can't</span> <span m="4900350">even</span> <span m="4900490">draw</span> <span m="4900750">because</span> <span m="4901010">it</span> <span m="4901110">gets</span> <span m="4901270">so</span> <span m="4901470">small</span> <span m="4901790">so</span> <span m="4901950">fast.</span> <span m="4903110">This</span> <span m="4903350">height</span> <span m="4905320">here</span> <span m="4906150">is</span> <span m="4907140">about</span> <span m="4907640">1</span> <span m="4907840">over</span> <span m="4908070">root</span> <span m="4908270">n.</span> <span m="4910840">And</span> <span m="4911040">this</span> <span m="4911230">width</span> <span m="4911820">here</span> <span m="4912390">of the</span> <span m="4912665">hump,</span> <span m="4913550">is</span> <span m="4913730">about</span> <span m="4914160">root</span> <span m="4914620">n.</span> <span m="4915920">And</span> <span m="4916070">these</span> <span m="4916320">things</span> <span m="4916910">zoom</span> <span m="4917850">down</span> <span m="4918110">to</span> <span m="4918170">0,</span> <span m="4919100">exponentially</span> <span m="4919720">fast.</span> </p>
<p><span m="4921470">And</span> <span m="4921660">so</span> <span m="4921770">that's</span> <span m="4921970">what</span> <span m="4922070">the</span> <span m="4922150">binomial</span> <span m="4922680">distribution</span> <span m="4923190">looks</span> <span m="4923500">like.</span> <span m="4924220">And</span> <span m="4924360">so</span> <span m="4924520">it</span> <span m="4924610">says</span> <span m="4925170">that</span> <span m="4925320">you</span> <span m="4926830">are</span> <span m="4926970">very</span> <span m="4927500">likely</span> <span m="4928350">to</span> <span m="4928460">be</span> <span m="4928580">very</span> <span m="4929080">close</span> <span m="4929650">to</span> <span m="4930860">pn</span> <span m="4932110">heads,</span> <span m="4932970">or</span> <span m="4933100">pn</span> <span m="4933550">things</span> <span m="4933800">happening.</span> <span m="4935640">And</span> <span m="4936010">I</span> <span m="4936050">won't</span> <span m="4936290">go</span> <span m="4936410">through</span> <span m="4936510">the</span> <span m="4936600">math</span> <span m="4936890">now--</span> <span m="4937190">probably</span> <span m="4937490">do</span> <span m="4937620">it in</span> <span m="4937700">recitation</span> <span m="4938250">tomorrow--</span> <span m="4939050">of</span> <span m="4939250">computing</span> <span m="4940300">at</span> <span m="4940470">most</span> <span m="4940790">25</span> <span m="4941270">heads,</span> <span m="4942160">which</span> <span m="4942400">is</span> <span m="4942480">the</span> <span m="4942570">cumulative</span> <span m="4943160">distribution</span> <span m="4943770">function.</span> </p>
<p><span m="4945310">And</span> <span m="4945430">this</span> <span m="4945580">comes</span> <span m="4945830">up</span> <span m="4945960">in</span> <span m="4946050">all</span> <span m="4946230">sorts</span> <span m="4946470">of</span> <span m="4946540">places.</span> <span m="4946930">Like</span> <span m="4947110">you</span> <span m="4947440">have</span> <span m="4947570">a</span> <span m="4947620">noisy</span> <span m="4948040">communications</span> <span m="4948760">channel,</span> <span m="4949920">and</span> <span m="4949980">every</span> <span m="4950230">bit</span> <span m="4950630">as</span> <span m="4950790">dropped</span> <span m="4951200">with</span> <span m="4951340">1%</span> <span m="4951930">chance.</span> <span m="4953620">If</span> <span m="4953750">you</span> <span m="4953840">have</span> <span m="4954030">10,000</span> <span m="4954720">bits,</span> <span m="4955010">you'd</span> <span m="4955150">like</span> <span m="4955340">to</span> <span m="4955430">know,</span> <span m="4956290">what's</span> <span m="4956640">the</span> <span m="4956730">probability</span> <span m="4957330">I</span> <span m="4957380">lost</span> <span m="4957880">2%</span> <span m="4958520">of</span> <span m="4958620">them</span> <span m="4960000">when</span> <span m="4960190">I</span> <span m="4960280">have</span> <span m="4960410">1%</span> <span m="4961490">failure</span> <span m="4961840">rate?</span> <span m="4962990">The</span> <span m="4963100">chance</span> <span m="4963450">of</span> <span m="4963560">losing</span> <span m="4963870">2%</span> <span m="4964630">out</span> <span m="4964760">of</span> <span m="4964840">10,000</span> <span m="4966050">is</span> <span m="4966190">like</span> <span m="4966430">2 to</span> <span m="4966650">the</span> <span m="4966750">minus</span> <span m="4967120">60</span> <span m="4968290">if</span> <span m="4968430">they're</span> <span m="4968540">mutually</span> <span m="4968890">independent.</span> <span m="4969500">So</span> <span m="4969580">no</span> <span m="4969820">chance</span> <span m="4970210">of</span> <span m="4970320">losing</span> <span m="4970620">2%.</span> <span m="4972140">All right,</span> <span m="4972340">very</span> <span m="4972780">good. We'll do</span> <span m="4973070">more</span> <span m="4973360">this in</span> <span m="4973570">recitation</span> <span m="4974180">tomorrow.</span> </p>
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